Non-Geometric Calabi-Yau Backgrounds and K3 Automorphisms C

Non-geometric Calabi-Yau backgrounds and K3 automorphisms C. Hull, D. Israël, A. Sarti

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C. Hull, D. Israël, A. Sarti. Non-geometric Calabi-Yau backgrounds and K3 automorphisms. Journal of High Energy Physics, Springer, 2017, pp.084. 10.1007/JHEP11(2017)084. hal-01651614

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Distributed under a Creative Commons Attribution| 4.0 International License JHEP11(2017)084 Springer Superstring October 16, 2017 November 5, 2017 : November 15, 2017 : y on mirror-folds : EUIL, France Received e are additional massless = 4 supergravity in four , f an automorphism of the ing theory which preserve Accepted , symmetric Gepner models. Published onodromies involving mirror N = 2 supergravity coupled to be the same four-dimensional f asymmetric Gepner models. 8, Universit´ede Poitiers, utomorphisms that arise from N Published for SISSA by https://doi.org/10.1007/JHEP11(2017)084 d [email protected] , and A. Sarti . 3 b,c 1710.00853 The Authors. Diﬀerential and Algebraic Geometry, Supergravity Models, c

D. Israël We consider compactifications of type IIA superstring theor , a [email protected] = 2 supersymmetry. The massless sector consists of E-mail: CNRS, UMR 7589, LPTHE, F-75005, Paris, France Laboratoire de et Mathématiques 2, Applications, UMRTéléport CNRS 734 Boulevard Marie et Pierre Curie, 86962 FUTUROSCOPE CHASSEN The Blackett Laboratory, Imperial CollegePrince London, Consort Road, London SW7LPTHE, 2AZ, UMR U.K. 7589, Sorbonne UPMC4 Universités, Univ. place Jussieu, Paris Paris, 06 France [email protected] b c d a Open Access Article funded by SCOAP Abstract: C.M. Hull, Non-geometric Calabi-Yau backgrounds and K3 automorphisms symmetries. At special pointsThe compactifications in are the constructed moduli fromthe space non-geometric diagonal these a action are of a anmirror automorphism surface. of the We K3 identify surface and the o corresponding gaugings of obtained as K3 fibrations over two-tori with non-geometric m In this way, we obtainN a class of Minkowski vacua of type II str dimensions, and show that thelow-energy minima physics of as the potential the descri worldsheet formulation in terms o Keywords: Vacua, Superstrings and Heterotic Strings ArXiv ePrint: 3 vector multiplets, givinghypermultiplets. the STU model. In some cases ther JHEP11(2017)084 7 8 1 6 30 31 34 35 10 15 18 21 23 29 29 30 eir 32 37 38 15 22 y transition func- : summary lattice sition functions are T-folds or ing backgrounds and have in- eaks some of the symmetries, e stringy duality symmetries. l for probing quantum geome- 20 , uch as asymmetric orbifolds can 4 Examples arise from spaces with – 1 – = 2 theory N 2 T ]. Such non-geometric spaces often have fewer moduli than th 1 ], while those with Calabi-Yau fibrations and mirror symmetr 4.2.1 Massless4.2.2 multiplets Massive4.2.3 multiplets Accidental4.2.4 massless multiplets Further accidental massless multiplets from KK modes 1 = 4 gauged supergravity from duality twists 5.1 Gravitini masses5.2 and supersymmetry Moduli space 4.2 Aspects of the low-energy 3.2 Symmetries of3.3 Landau-Ginzburg mirror pairs Worldsheet construction of non-geometric backgrounds 4.1 Twisted reduction on 2.1 Lattice-polarized2.2 mirror symmetry mirror Berglund-Hübsch symmetry 2.3 Non-symplectic automorphisms of prime order 3.1 Mirrored automorphisms and isometries of the Γ N tions are mirror-folds [ geometric counterparts, and theincluding non-geometry supersymmetries, typically and br providetry. an Solvable worldsheet interesting conformal too field theories (CFT’s) s U-folds [ 1 Introduction Geometric compactifications constitute onlyteresting a generalisations subset to of non-geometriclocal str backgrounds. fibrations thatSpaces have with transition torus fibrations functions and that T-duality or includ U-duality tran A Explicit lattice computations 5 Compactifications with non-geometric monodromies 6 Conclusion 4 3 Non-geometric automorphisms of K3 sigma-models Contents 1 Introduction 2 Mathematical background JHEP11(2017)084 d ). T Z , ) must i the ansatz πR Schwarz re- ]. From the ], allowing a 2 gφ 2 [ ) monodromy + 2 i Z l see, particular ( y 7→ ken to a discrete ( of the K3 surface. 1 dimensions and G g φ 1 1, are the remaining − πR/r − as tions with a duality ) − dromies. For example, i D d y G ( − ring constructed from K3 g ty twist can be viewed as a has a minimum at which it etries, i.e. which preserve a try; in the present work, we re points in the moduli space xtends to a reduction on c background [ lds in s. point in moduli space (i.e. at ant under finite subgroups of ). half of the 16 supersymmetries Z nd a supergravity construction, circles. The question of finding ( -torus with a ,...,D d d G ) (1.1) = 0 ] K3 mirror-folds with monodromies x µ 3 ( ˆ , φ ) µ ]. In these cases, at the special points in y dimensions contains the moduli space for x 2 ( dimensions with discrete duality symmetry transforming under g , the mondromies dimensions has an action of the continuous i D φ D ] at low energies. dimensions becomes related to that of finding and 4 D πR ) = d – 2 – d i T − 6. If one considers reductions on a circle where the , y + 2 , µ i D 4 x , y ( 3 φ , ∼ i subgroups, then at any point in the moduli space invariant y 3. As a result, we find interesting mirror-folds which give = 2 r r Z K for r U(1), identified under the action of the discrete group SL(2 / Z circles, giving a string-theory generalization of Scherk- ) . , are coordinates on R twist together with a shift around the circle by 2 i d , r Z with transition functions involving the mirror involution 2 , . . . , d T = 1 = 2 Minkowski vacua of type IIA superstring theory. As we shal ]. In many cases, the theory in ) for each i dimensions with a duality twist on each of the which is a symmetry of the low energy physics, but which is bro 5 Z , i ( d N ]. In such a construction, a theory in ) isomorphic to G y 2 G Z − dimensions that happen to be fixed under the action of the mono , Our focus here will be on mirror-folds of the type IIA superst Our constructions can be viewed as particular cases of reduc Of particular interest are the special cases in which there a ) (e.g. T-duality or U-duality) is compactified on a orbifold by a D Z = 4, D ( r bundles over to fixed points in moduli space preserved by a subgroup of reductions giving Minkowski space in Previously Kawai et Sugawara have considered in [ D duction [ giving complementary pictures of the same reduction. This e effective field theory point of view, the moduli give scalar fie moduli space there is both a stringy orbifold construction a twist [ G consider in contrast monodromiesquarter that of the preserve 32 8 supersymmetries of supersymm of the type type IIA string, IIA or compactified a on complete analysis and important checks on general argument that, at least when the fiber is compact, break all supersymme around each of the the reduction gives a potentialeach for point these that fields. is At preservedvanishes, each by giving fixed the a monodromy) Minkowski the compactification potential [ examples give precisely the STU model of [ subgroup in the full stringis theory. of For a the field form arise at special points in the moduli space of a non-geometri group where be in under the action of theZ monodromy, the reduction with a duali coordinates. With periodicities in consider a theory wherea the 2-torus, moduli space i.e. in SL(2 monodromy is in one of these There are special points inSL(2 the moduli space which are invari JHEP11(2017)084 , 20 , 4 con- e su- 3 gives iew, we K without twists 2 ] and explored by T 8 -fields on ], following earlier nspired by a world- 9 B , rlie these compactifica- 8 ell as the lattice of D- ned from compactifying tion on otential, and of particular n [ y compactifications whose ma-models with K3 target truction of the low-energy on a class of monodromies = 2 supersymmetry. s that for our constructions, han being tied to the string uthors in [ cal framework – that applies f the low-energy physics. We try means that they are non- ty with a non-abelian gauge N f (spontaneous) space-time su- ations (rather than from iden- ies) and analyse the worldsheet eometric automorphisms, rather ), and taking orbifolds by such 20) preserving the lattice Γ ima that give zero vacuum energy , 20 , ving rise to fibrations over tori that 4 = 4 abelian vector multiplets. The O(4 , ] N 7 ) twist around each circle. Generically, ⊂ , 20 6 , ) 4 20 , 20) O(20) 4 , × – 3 – = 2 supersymmetry from an algebraic geometry O(4 N O(4) ) is the perturbative duality group of two-dimensional with an O(Γ 20 , 2 4 3 target spaces, which includes mirror symmetries. We then T K ] for later generalizations). They preserve only space-tim ] for a related but distinct class of models. In that work they 3 surface. The moduli space of metrics and 13 K 12 , 3 CFT’s and is given by [ K 12 3 mirror-folds with K = 2 vacua in four dimensions. From the string theory point of v ] (see [ 11 N , = 4 supergravity coupled to twenty-two 10 N The particular automorphisms of K3 CFTs that we will use are i The relation between the asymmetric Gepner models that unde From the supergravity point of view, the conventional reduc Our starting point will be the theory in six dimensions obtai sheet construction of asymmetric Gepner models presented i admitting will find them byspaces considering points that in are the preserved moduli by space finite of subgroups sig of O(Γ such reductions break all supersymmetry; we will focus here automorphisms combined with shifts on the circles. supergravity limits are of this type, and moreover preserve twisted reduction gives agroup and gauged a version scalar ofinterest potential. are this theories Vacua with arise supergravi non-negative from potentialsand minima and min a of the Minkowski p vacuum. Our construction gives string theor tions and gauged supergravities was suggested by one of the a gives viewpoint. This approach willto provide Calabi-Yau a powerful three-folds mathemati asfour-dimensional gauged well supergravity. – and give an explicit cons reduce to four dimensions on which is thebrane lattice charges of in total type cohomology IIA. of O(Γ the K3 surface as w tifying the massless spectraconstructions of of the four-dimensional theor than the freely-acting quotients involving toruswe shifts consider gi here. Inpersymmetry the breaking freely-acting can quotient, bescale the made as scale arbitrarily it o small would be ratherthe for t gauged quotients supergravity with approach fixed points. giveswill a This identify good mean the description gauging o directly from geometric consider Blumenhagen et al. in [ sidered quotients of Calabi-Yau compactifications by non-g works [ percharges from the worldsheet left-movers,geometric and in the general. asymme IIA string theory on a conformal field theories with identified under the discrete subgroup O(Γ the moduli space of JHEP11(2017)084 , T p ⊕ ρ σ r.t. ) can that p (1.3) (1.2) nwall σ 20 , ( 20 , 4 4 T notion of phism ]; see the ar- 17 . These surfaces – n 15 to physicists, is the . The first factor is ,...,w . As we will review in 2 1 le than for Calabi-Yau ). It is expected that a w olution of) the zero-loci sociated with the action T p P tion of the automorphism σ iπ/p . They can be decomposed ( tion p T er, and we expect the physics ) ebraic surface and the second ct on sub-lattices of Γ 3 mirror symmetry, which will ρ paces c automorphisms, is to consider , coincides with the Berglund- ) K p , the rank of their Picard lattice O( = exp 2 ns (BH and LP mirror symmetry) , σ ρ , interchanging the two factors in ) and , ρ p p p ρ × ζ σ σ ( − ◦ O(2 T µ O(2) ◦ ) transformation that maps a K3 surface × T p 20. For the algebraic K3 surfaces of fixed 3 CFTs of order 20 σ ) , 3 CFTs that we study in this work correspond 4 K ρ 6 ◦ – 4 – ) , where e.g. K ρ 1 ] to generic Landau-Ginzburg models with non- of − ρ ω − Picard number 20 6 p µ − ζ := O(20 20 7→ , ], which are automorphisms of the surface acting on the (LP-mirror symmetry) was introduced around 40 years p ] (BH mirror symmetry), generalizing the Greene-Plesser × ˆ ω σ ]. An important corollary that we will obtain is that the 21 [ 19 : O(2 23 ) where 1 ⋆ , p O(2) σ X 22 ( is a prime number, the lattice-polarized mirror symmetry w. 1) as on the mirror K3 surface. These isometries of the lattice Γ p , T (1 p ω ] for an introduction. In the special case considered by Aspi σ ] and independently by Dolgachev and Nikulin [ to one with Picard number 20 H 18 ) transformation induced by a block-diagonal isometry in O( ρ ∩ 14 of the surface on the one-hand and the corresponding automor 3 is a manifold. hyperkähler For algebraic K3 surfaces, the 20 ) , ], this amounts to an O(Γ mirrored automorphisms p 4 Z 7 K σ X, ( 2 H , at least when 2 ). Another notion of mirror symmetry, perhaps more familiar ) = )), the first block giving the isometry associated with the ac 1.2 A key ingredient to reconcile these two approaches to The definition of mirror symmetry for K3 surfaces is more subt The non-geometric automorphisms of We consider algebraic K3 surfaces defined as (the minimal res p on the K3 surface, and the second block giving the isometry as X σ ( ( p σ lattice-polarized mirror symmetry ago by Pinkham [ T also play a central role innon-symplectic the automorphisms present study of non-geometri of Picard number eq. ( three-folds, as section each to an O(Γ automorphism overlap but do not always agree. holomorphic two-form egudHushconstructionBerglund-Hübsch [ construction of mirror Gepnerdegenerate models invertible [ polynomials. These two constructio ticle by Dolgachev [ and Morrison [ interpreted as the complex structureas moduli the space complex moduli of Kähler space. the alg of quasi-homogeneous polynomials in weighted projective s of the automorphism are characterized in particular by their identified under a discrete subgroup, as we will review in sec be thought of as are orthogonal to each other, denoted respectively of the mirror (Berglund-Hübsch) surface on the other hand a similar statement is true forto automorphisms work of out non prime in ord a similar fashion for such cases as well. as follows: ¨bc mirrorHübsch symmetry [ S the invariant sublattice associated with the action of the moduli space of CFTs factorizes into JHEP11(2017)084 , 3 K 3.2 large ]. At 1 p ) mon- li space ensional )[ 20 , 4 20 , 3 to obtain 4 ctions of the ) acts on the K that the fixed 20 ) monodromies , 4 3 20 , is an order shifts on the two 4 p with O(Γ on σ large diffeomorphism up to subsection 2 presents the mirrored tively dual to the het- O(Γ p T 3 wists. It is in general a Here otient by elements of the ric orbifold type. Finding is often referred to as the d precisely to the Gepner space fixed under the two s, giving vacua preserving ifficult problem; the novelty recisely the type introduced , gives a rich class of models ticle deals with the physical s on f the twists, the construction ompactified on g class of automorphisms that, ]. We extend this construction the moduli space these become ntaining T-duality transforma- 8 T-duality group O(Γ ]. Then the type IIA K3 mirror- is an order 2 [ T p 2 σ T 3 back to the original one. K – 5 – 3 to its mirror, ]. The duality symmetry group O(Γ 3 surfaces and their relation with asymmetric Gepner K 3 automorphisms and section 24 K [ , we will consider the Scherk-Schwarz compactification of K 4 ) transformations that have fixed points in the K3 moduli 4 = 4 supergravity. We show that for suitable choices of the 20 T , provides the necessary mathematical background about 4 N 2 maps the mirror 1 maps the ] on the worldsheet (which are Landau-Ginzburg points in the ) combined with shifts on − 8 µ µ 20 , 4 3, into the gauged supergravity framework, and obtain the modu K 3 CFTs with enhanced discrete symmetry) and lead to four-dim 3 ]. These reductions can be regarded in general as T-fold redu = 2 Minkowski vacua. K 3, and 2 N K denotes the mirror Berglund-Hübsch/lattice involution. = 2 supersymmetry. µ N The type IIA string theory compactified on K3 is non-perturba This work is organized as follows. The first half of the paper, Taking the quotient by two such automorphisms combined with = 2 supersymmetry. In the final section, we will translate the defined in section erotic string compactified on tions as well asheterotic diffeomorphisms T-duality and group. shifts of The the reduction B-field, from and 6-dimension heterotic side through isometries of the Narain lattice, co diffeomorphism of a four-dimensional gauged twists, two gravitini becomeN massive and two remain massles theories with odromies round each circleand studied provides in twisted [ reductions of p one-cycles of a two-torus,automorphisms, at gives the special asymmetric points Gepner in models the of K3 [ moduli of the mirror model constructions. In section of the low-energy theory. is more algebraic-geometry orientedaspects. while In the detail, rest section of the ar automorphisms of mirror pairs of the 6-dimensional supergravity corresponding to type IIA c here is that algebraic geometrywhen leads used us in to either awith the very type interestin IIA or the heterotic description the special points ofreduces the moduli to space a that reductionT-duality are fixed-points group of o O(Γ asymmetric orbifoldfolds are type, dual with to a heterotictype T-folds, qu IIA and Gepner-type at models specialautomorphisms and points with in heterotic interesting models fixed of points asymmet is in general a d moduli space of to all pointsdifficult in problem moduli to find space O(Γ using a reductionpoints with of duality the t monodromiesmodel that construction we of consider [ correspond indee space and so can lead to Minkowski vacua. We will show in secti where surfaces, mirror symmetry and heterotic string, with transition functions involving the JHEP11(2017)084 1 is . 8 8 (2.6) (2.5) (2.7) (2.4) (2.2) (2.1) (2.3) E gonal E . 1). The )) ), gener- endowed ρ 20 − , X ), or Picard 4 O( 1) and dual) lattice , ρ , X × Picard lattice ( ρ O(Γ lattice. The geo- ⊂ . The moduli space (O(2) 20 , is the ⊥ / 19 4 , ) ) ) and transformations 3 X ttice can be enlarged to X , ρ 20 , ( 4 , Q ⋉ Z integral cohomolgy classes. ignature is (1 . ) S O(2 h a Picard lattice of Picard \ s defined by quantizing non- 19 ) of a K3 surface ∩ 19 O(Γ , belian group) . , 3 )) Z 3 is defined to be the sub-lattice ) 20 O(20) Γ ⊂ , U, X 4 X ( X, ) X × ( curves of the surface, i.e. curves that are ( ⊕ → Q ֒ 2 19 1) , , S U 3 U. ⊥ H (1 ]: ) = Γ ) of a K3 surface U, O( 7 O(4) ⊕ ⊕ H / X X , ) ⊕ 19 ( ( × U , 6 ∩ 3 X T S ) 19 20) ( )) ⊕ , , 3 Z ρ ∩ 8 S 8) associated with the Dynkin diagram Γ ) − are ‘non-geometric’. , E – 6 – X, ∼ = Z O(4 ∼ = ( ⊕ 19 \ ) 2 , ) 8 3 20 X, , X O(20 H ( 20 4 E ( , 2 × 4 Γ Q ). ⋉ Z ∼ = H ρ S ) = ) 19 O(Γ − X 19 , , ( 3 ) = is the even unimodular lattice of signature (1 (O(2) 3 ]: ∼ = Γ / 20 S 7 X ) , U ( Σ ρ . ) of an algebraic K3 surface T 3 CFT’s correspond to isometries of the Γ − X M X ( K 20 , T 20): orthogonal to the Picard lattice: , ). The transcendental lattice can also be viewed as the ortho O(2 19 , ρ \ , 3 )) Γ ) is the isometry group of the even and unimodular (i.e. self- X ( ∼ = 20 , ) T 4 then factorizes as [ is the K3 lattice Z ρ O( 19 , X, ∼ = 3 ( of signature (4 corresponding to shifts of the B-field by representatives of 2 For sigma-model CFTs on algebraic K3 surfaces, the Picard la Automorphisms of An important sublattice of the K3 lattice Γ ρ Σ The Picard group or Picard lattice is generated by algebraic 19 20 1 H , , 3 4 M complement of the quantum Picard lattice, i.e. number of non-linear sigma-model CFTs on algebraic K3 surfaces wit with signature (2 This lattice has signature (2 the ‘quantum Picard lattice’: transcendental lattice of number, is at least one for any algebraic K3 surface, and its s The rank of this lattice (i.e. the rank of the corresponding A holomorphically embedded in which is defined to be: The moduli space oflinear two-dimensional sigma conformal models field on theorie K3 surfaces is given by [ 2 Mathematical background and Γ Γ which is isometric to the second cohomology group where O(Γ with its cup product. Here the even unimodular lattice of signature (0 Isometries that are not in O(Γ metric automorphisms of the surface form a subgroup O(Γ ated by large diffeomorphisms of the surface in O(Γ Z JHEP11(2017)084 , , ). X ⊥ 18 ) X and as a , the ( (2.8) (2.9) (LP) } M 1 3 sur- 6 2 S , ( e 2 ι t M K h − ], Nikulin -polarized ) = 1 → ∨ 14 , e M 3 surface 2 M ( -polarized K3 ֒ U e ι , with K ∨ ) ] is a particular is 2 + : 7 , t M 1 ∨ ′ e that we will review ι (1 { X . Assuming that its is defined through the lattice-polarized -polarized K3 surface 19 gebraic , ) 3 t duli spaces. For M as an Abelian group and Hodge numbers ion of the non-geometric Γ ex structure moduli space − in the Picard lattice of an ). The moduli space of the mbiguously defined. For al- N → 18 M , , M e. A primitive embedding of a ∨ (1 ֒ M rk ( of ) is spanned by and signature M -polarized and . : ) − M ) as these manifolds are hyperkähler, ∨ -polarized K3 surface. Then two K3 ( ⊕ M ι X ι ii ) ( + 1 M M 3 surfaces is the is embedded primitively as S t U ( ⊕ K ′ . X ι ) ) = 20 ] and introduced by Pinkham [ 2 M → ∨ Z U , and if is an 18 ( ′ ∼ = 3 surfaces are diffeomorphic to each other and ) = M N . ι and signature ֒ M ) – 7 – X K X t ) is a torsion-free Abelian group. = ( M : admits a primitive embedding ( , is such that, viewing − T ⊥ M  ) ( N ) all 19 = N/ι , 19 i 3 M ⊥ ( → Γ N/ι ) ι . X ⊂ ֒ M ( ) is a basis of M 2 S ⊥ : of rank ) , e ι 1 e , M M ( ι N , then we say that be an even lattice of rank to ]. This construction uses the embedding of a given lattice X is not primitive as ∨ form a lattice mirror pair if 17 M LP mirror moduli spaces – N M ∨ 15 → X Let The Aspinwall-Morrison construction of mirror symmetry [ surface ֒ M -polarized K3 surface, with rk( -polarized K3 surfaces and the moduli space of the family of 3 and : ∨ into a lattice 3 surfaces there are two different notions of mirror symmetry ι the mirror lattice to K M X M K ∨ M M In other words, the LP mirror construction associates each Mirror symmetry of Calabi-Yau three-folds exchanges their ) as a subgroup, the quotient , and exchanges the complex structure and complex mo Kähler As a counter-example, if ( 1 2 , M 1 ( surfaces are called with an family of Example 1. with A first notion of mirror symmetry of algebraic the complex structure and complexgebraic moduli Kähler are not una in turn below, andautomorphisms we these use will in both this paper. play a role2.1 in the construct Lattice-polarized mirror symmetry algebraic surfaces mirror symmetry discussed by Dolgachev in [ the mirror lattice If there exists a primitive embedding decomposition orthogonal complement so have the same Hodge numbers and topology, and ( admitting a primitive embedding in the K3 lattice, faces the situation is different since ( lattice sub-lattice of the Picard lattice, which should be primitiv instance of LP mirror symmetry. The authors considered an al and Dolgachev [ embedding polarized by the whole Picard lattice, i.e. with Definition 1. and the second one to the complex moduli Kähler space. One may think of the first factor as corresponding to the compl h One has then ι JHEP11(2017)084 . → 4 C (2.13) (2.12) (2.11) (2.15) (2.14) (2.10) , is not : reserving lund and invertible W ] of mirror exchanging -currents of , 20 R } ) 4 irror symmetry is . , the hypersurface ) d . If furthermore the 4 , . . . x , 1 ] that it coincides with W = Z x ( ℓ on of a hypersurface in a 27 , ∈ onstruction [ w ) dental lattice. Hence both W ℓ , . . . , x ∨ omial is said to be g of sigma-model CFTs on the 1 =1 4 ℓ X x in the form 4 ) = ( =1 ( ℓ ) = 1. A polynomial X 4 which is a smooth K3 surface. P T 4 x W W 4 d W , , λ ) = X j i = 0 is the only critical point and if 4 a X , . . . , w ( , . . . µ ) = 4 j , 1 iπg 1 4 Q x x 2 ∨ x w x 1 4 X 4 = µ =1 w Y j ( – 8 – 7→ , . . . , e 4 W =1 ··· ,S egudHush(BH)Berglund-Hübsch mirror symmetry 1 i X ) | X ∨ , . . . , λ 4 iπg = are uniquely determined by = ) is invertible. If ) 1 2 with gcd ( j X i ⋆ e x ] 1 ( if a 1 4 C W /d x Q ( w w d 4 3 S λ ∈ w ( ) = := ( 2 4 ) w 4 ) = 1 ]; later Chiodo and Ruan proved in [ W w X [ ( A ,W 26 P /d, . . . , w ∗ T , . . . , µ 1 admits a minimal resolution C 1 , . . . , µ ] the Abelian group of all diagonal scaling transformations p w 1 4 µ ∈ ( µ w W ( 3 λ ] discovered in physics, exchanging the vector and axial { 2) superconformal field theories. It was generalized by Berg w G : ∀ , 2 25 = w W 1 w ) are exchanged under this involution, which can be viewed as W [ P G 2.7 ] and Krawitz [ its subgroup containing elements of the form in is the Picard lattice of the mirror surface. In other words, m 19 } W ∨ M SL We denote by Let us consider a K3 surface realized as the minimal resoluti = 0 is quasi-homogeneous of degree W the fractional weights { where the square matrix number of monomials equals the number of variables the polyn It is non-degenerate if the origin C which exchanges the quantumfactors Picard in lattice ( and the transcen the polynomial By rescaling one can then put an invertible polynomial weighted projective space and where ¨bc [ Hübsch cohomological mirror symmetry. specific to K3Gepner surfaces. models [ It follows from the Greene-Plesser c defined as the map the worldsheet (2 and 2.2 mirror Berglund-Hübsch symmetry The second notion of mirror symmetry, the complex structure and thesurface. complex moduli Kähler spaces JHEP11(2017)084 . 3 h i × = 7→ ) 3 K ℓ h /d x 4 (2.17) (2.16) d/w Z , is also × , a smooth 2 W,G ˜ fined by the 4 while G X , / ) generating a d/w 3 /d, . . . , w , W Z 1 /d 2 w 4 X , × ( , and the quotient , the minimal reso- w h 1 ) is associated with T W = 1 = = , then the mirror pair ,G d/w ) is an automorphism 4 ˜ ℓ 4 , denoted T satisfying the condition SL Z G W ch follows in the physical ˜ W, G / G W J G ∼ = / ⊆ , ld theories associated with W for X preserving symmetries. The 4 W ℓ W X G . , . . . , g x X 1 G } and the other coordinates are d/w ) ˜ to be 4 g /d, . . . , g 1 ⊆ W inal Greene-Plesser construction ℓ 1 x ℓ G G − x w is given by the elements W + ) = ( ). In physics, an orbifold of a πi g J = 3 T = /d g T ℓ . 1 G d/w g T 3 W ) W, G , . . . , d x exp (2 G 0 πi w W under which the coordinates scale as . + is then given by ∈ } . The mirror surface is characterized by the A 7→ 2 ∈ { ) W with ( G W 4 such that ℓ r – 9 – X 3 surface obtained as the minimal resolution of W,G x d/w ∈ X := ( 2 ∀ exp (2 W , W ) = 0 and x j X K h T 4 = as ˜ Z ∀ G ˜ x + 7→ W , . . . , g ℓ , 1 ∈ 1 A ℓ x ⊂ g ℓ W,G Z x . ,... g d/w 1 G X 1 ∈ is obtained as follows: T = ( x x 4 =1 . ) (˜ W T ℓ X g , and the dual group T T = T h r f G SL W W be associated with the minimal resolution of W ,G . / W ). This polynomial is preserved by the symmetries under whic d ) T T 4 ⊆ = W T T W , gA ( X G T W, G W ( W ⊆ , . . . , w . The minimal resolution of the orbifold 1 preserves all space-time supersymmetry. G T of order W w Let W ∈ W A Fermat-type K3 surface in a weighted projective space is de W . It is straightforward to check that, if the pair ( J ℓ g J 3 surface obtained as the minimal resolution of { is the surface x G/J always contains the element SL ) ) specifies an automorphism of is specified by the matrix K ℓ 4 = T , as = ) is associated with a smooth acting on the coordinates ˜ ⊆ is the trivial group and = lcm ( W T T W T T ˜ f d T G ˜ G G G G πi h W G X . Consider the case in which we choose the group / ,G W 4 We now introduce mirror the symmetry, Berglund-Hübsch whi Let us consider a subgroup ⊆ T • • T , . . . , h X 1 W d/w W W h satisfying the condition Then Z same polynomial as invariant. They generate the group of diagonal symmetries X Example 2. polynomial a smooth ( any of the coordinates scales as where exp (2 ( Definition 2. context from the isomorphisma between pair the of superconformal Landau-Ginzburg orbifolds,of fie generalizing mirror the Gepner orig model orbifolds. lution of the orbifold sigma-model (or Landau-Ginzburg model)J by a discrete group K3 surface. The pair a K3 surface, that we associate with the pair ( group of Here cyclic group The mirror Berglund-Hübsch surface of group which corresponds in physics to the group of supersymmetry- JHEP11(2017)084 as a ) as (2.19) (2.21) (2.18) can be X . There X ( C X ω ⊗ ) ) be the sub- p p σ ) its orthogonal σ ( p ( 3 surface σ T S ( K but for simplicity well T . As was shown by p ρ of a . Let and rface. There exists nev- ⋆ p p ⋆ p σ σ sults as the Berglund-Hübsch 3 lattice. The following lemma          tion of non-geometric automor- q generates a cyclic group isomor- K I in which the surface is polarized p 1 , ...... ζ 0 0 − ) p p ζ X purely non-symplectic 3 surfaces do not necessarily agree. In . ( on the vector space . ) (2.20) ω . Z 0 K ⋆ p . p X σ ( ζ ∈ q S ℓ I 7→ n n p ) will be denoted by ⊆ ζ p ) ℓ ) d is a prime number then it is straightforward to σ is a primer number, as it is the case in the present work, – 10 – ) is a subset of the Picard lattice, w p X ( p ( p p σ ········· S σ ( 4 ω =1 ( . If ℓ acts on 2-forms through X S . S 0 : . ········· 3 ⋆ X p ). q non-symplectic automorphism of a K3 surface I σ ...... , we get the condition 0 0 p ℓ → p ζ ], the LP mirror of a surface polarized by its whole Picard n iπ/p          in this case. X 28 of the surface acting on the holomorphic two-form ], Theorem 0.1). : such that the action W X p 21 ) are primitive sub-lattices of the q σ -th root of unity, i.e. such that p SL p → σ ]: = exp(2 ( = X p T 21 be an order : ζ T ) invariant under the action of the isometry p p G 19 (see [ Z σ σ ≤ ) and for some integer X, , e.g. p p ( Let ℓ σ Z 2 d ], the invariant sublattice ( w ≤ ℓ S H /p is a primitive 21 n Z p ζ = ℓ Let us first define a non-symplectic automorphism of order The automorphism These autormorphisms are often called in the literature g 3 phic to where diffeomorphism was also proved in [ particular, as was shown in [ 2.3 Non-symplectic automorphismsThe of two prime notions order of mirror symmetry of algebraic This means that diagonalized as and both lattice of Lemma 1. by a sub-lattice ofconstruction, the and Picard that lattice, will that bephisms. gives instrumental the in same our re construc lattice is not alwaysertheless identical a to class the of lattice-polarized BH mirror mirror symmetries, of the same su see that 2 complement. The rank of the lattice As Nikulin [ exists a positive integer call them just non-symplectic. When the order each non-symplectic automorphism is purely non-symplectic. JHEP11(2017)084 : ∈ is 20 ) = ) , p with non- 4 T p X (2.23) (2.22) (2.24) Γ σ -cyclic ( } p ( p 1 S in T irror fol- is the in- ) − ⊆ ) p p ) σ me order up to -cyclic form, as p , p ( σ p its LP mirror, the real vector ( ] and for 1 σ surfaces in the T ( S R T 3 22 S of L ,G ), therefore it also ∈ { K 5 4 T ˜ x n W , and similarly 3 y an important role in be its Berglund-Hübsch X ed projective space ad- ˜ x ) , T T p 2 ometries. σ ,G x lattice, namely ( (˜ T ˜ S . and f 20 W } , st define the quantum invariant ⊕ we denote by 4 + X tion of the non-symplectic auto- 13 . , whose non-degenerate invertible p U L . 1 , p 7 ) x 20 = , , 4 4 5 U, = ˜ . Let , Γ , x 19 associated with the non-symplectic au- , } 3 3 ⊕ T 3 ) , ] the action of the non-symplectic auto- ∩ but we tacitly choose the embedding such that ) ]. They admit the obvious order ) in the Γ Γ 13 2 T , x . p p W p ⊥ , 21 2 = 2 by Artebani et al. [ ˜ 20 x ∩ ) σ σ 23 , 7 σ x ( 4 ( p p ( , -roots of unity are eigenvalues and the corre- ∈ { ⊥ ( one cannot present the surface in a p ζ -polarized K3 surface, where σ S 5 ) S p T f -polarized K3 surface, where ) ( p . } p , belong to mirror families of . By construction, the BH mirror of a ) p q 3 is the orthogonal complement σ 1 ∼ = T – 11 – p 7→ + 19 σ ). For a lattice , ( T x , σ ) ( ) 2 p x 1 ( S ∼ = p p X T ,G dimensions and all integers S p 17 x ( ζ σ S ) T , : ˜ σ ( ρ ∈ { q T ( p = W 11 Q 4 ) = T 7→ p σ T p 6 p , following [ ]. S X ( σ 1 σ p ∈ { W ( T x 23 ρ -cyclic p action, with -cyclic T p : p and p σ p p-cyclic . Hence σ ) be a ) with W,G be a p p ], K3 surfaces admitting non-symplectic automorphisms of pri ) may not be unique in Γ σ X ], Proposition 9.3) and [ p ( 21 , ρ σ -polarized K3 surface, which is also its m Berglund-Hübsch ( W,G S 29 ) W,G T T p X ⊕ automorphism X σ . U ( p 1 . Then S ]. Let = Let T appear once, i.e. all primitive p of prime order, with 23 From ([ generated by its basis vectors. σ 19 p by Comparin et al. [ , 3 σ R is the identity matrix in } Γ ) = 1 ⊗ q ∩ 13 I , L ⊥ 7 ) n, p ) is the orthogonal complement. , We obtain from this theorem a simple corollary which will pla From the theorem we have We will consider a particular type of hypersurface in weight The following theorem was proved for T p As shown by Nikulin [ The embedding of T 5 p 4 5 σ , σ = 19 exist; however for the values ( ( 3 lowing Theorem space regarded as an Corollary 1. which has signature (2 variant sublattice under the the construction of non-geometric compactifications.sublattice We as fir the orthogonal complement of sense of lattice-polarized mirror symmetry. tomorphism was noticed in [ Remark 1. { T p mitting a non-symplectic automorphism of prime order morphism on the K3 lattice is unique up to conjugation with is We will call such surfaces symplectic automorphism mirror, polarized by the invariant sublattice surface has its defining polynomial of the form polynomial is of the form Theorem 1. where gcd ( sponding eigenspaces are all of dimension S the sub-lattice of themorphism Picard lattice invariant under the ac admits an order JHEP11(2017)084 ] ) 7 er. 28 X , σ ( 3 S (2.27) (2.30) (2.28) (2.25) , σ 2 σ ), quotiented X is naturally the ( 9), which is ) of Fermat type. S , 20 , 4 ], Theorem 0.1), that 2.26 er first the self-mirror surface corresponds to 21 has Picard lattice of rank er to obtain a smooth K3 ism of order 42, acting on 1] rface ( , f the K3 lattice, the rank is 6 hen 10. Interestingly, by [ Morrison, as we have , ler function of 42, which is 12. O(10) (2.29) 14 , ies, by ([ U. . ). × [21 R ⊕ ) P X p ( U σ = 0 (2.26) : T ( x , y . O(2) R ⊕ 3 7 / U. T 42 ∼ = z is identified with the Picard lattice 8 z ) iπ/ iπ/ ⊕ w , 42 ⊕ E p 2 2 10) 19 + 8 , e e , R σ 3 iπ/ ) 7 ∼ = ( E 2 Γ y T p ) . It inherits the orbifold singularities of the T e 7→ − 7→ 7→ ∼ = σ O(2 1] – 12 – ⊂ + , X ( y x \ ) w 6 ( 7→ , ) 3 7 and the corresponding automorphisms T )) : : p , orthogonal to the basis vectors of T : x X z 14 ( , 3 σ 7 X 3 ∼ = ( , 2 ∼ = + 19 ( S σ , σ [21 σ S ). Its orthogonal complement in Γ 3 2 ) T P 20 , R w = 2 X X R 4 ( O( ( p Γ Q Q ∼ = S S cs ) admits several non-symplectic automorphisms of prime ord ∼ = ) M p ) is isometric to a self dual lattice of signature (1 2.26 σ ( X -cyclic for ( Q p S S given by the hypersurface In order to illustrate this general construction, we consid X ]), and )), the group of isometries of the transcendental lattice: 29 X , ( T 23 The moduli space of complex structures associated with this The hypersurface ( by transcendental lattice of the surface, hence the set of space-like two-planes in ambient space, and thesesurface. should This be K3 minimallyz surface resolved admits in a ord non–symplectic automorph Thus this surface is its own mirror in the sense of Aspinwall- 10, which is the same as the rank of the Picard lattice of the su and leaving the other coordinates invariant. Then this impl The hypersurface is by O( the generic K3 surface in the weighted projective space The Picard lattice the rank of the transcendentalSince lattice the is rank a multiple is ofnecessarily necessarily the equal less Eu to than 12 22, and which the is rank the of rank the o Picard lattice is t in the weighted projective space In all cases, the invariant sublattice are of order 2, 3 and 7. Their action is K3 surface We obtain then the orthogonal decomposition over of real dimension 20. Example 3. (see [ JHEP11(2017)084 ): U U ⊕ ⊕ (2.32) (2.31) (2.33) U U ) is given, is the matrix n x 2.26 + 1 − n lattice. For x 8 1 c is − E n a , + ): . 8 matrix) is               ··· 2 E               + − 1 2 3 4 5 2 3 2 x 1 . ) of the surface ( 2 1 − − − − − − − − a − +      0 2 0 0 2.28 1 1 1 2 1 2 1 3 1 4 1 2 1 2 1 1 a 6 2 0 . − − − − − − − − − . 1 − 2 1 1 0 ) = 1 2 1 1 x      1 0 1 2 0 3 . − ( − − − − lattice and onto the P − − P 2 1 0 0 0 1 0 1 0 1 U 3 0 − − − − 0 1 00 1 0 − ⊕ 2 1 0 0 0 0 – 13 –      − 0 1 01 0 0 00 0 0 00 1 0 1 0 1 1 1 1 1 2 2 2 0 1 1 0 2 2 0 1 1 0 0 U is then given by the following matrix in O( 2 1 0 0 0 0 0 = − − − − − − −      U − 0 0 0 0 0 1 2 2 3 2 2 1 U ⊕ ⊕ 2 1 0 0 0 0 0 0 ), using Lemma               0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 U 3 3 − U , by six copies of the companion matrix of the cyclotomic :=               M on iπn/ C 8 the action can be given in a unique way on the transcendental 2 E 3 3 := e 1 σ M st 8 − E z ( is given by the following matrix in O( on the transcendental lattice ( 8 =1 3 2 n E σ Q on , whose characteristic polynomial is =         3 1 3 σ 0 1 2 − . . . a a a n lattice, choosing a lattice basis in which the lattice metri a − − − 8 − . . . E 0 0 0 1 . . . ··· ··· ··· ··· Recall that by Remark The action of The companion matrix of a polynomial of the form . . . 6 . . . 0 0 1 0 0 1 0 0 in the appropriate basispolynomial over Φ lattice. It splits into an action onto the the action of         For the The action of the isometry we choose a lattice basis in which the lattice metric (or Gram JHEP11(2017)084 . , ], 8. ⊕ U U 6 23 ≥ ⊕ ⊕ E ) (2.36) (2.35) U U sions. X ⊕ ( 2 ) = is its own ). Second, S lattice, by T 3 A W σ 2 ⊕ ( 8 A 2.31 T W,J E X ), and finally the and the transcen- 7 )-polarized surface and ) = by two copies of the T 3 U 3 σ U σ iπn/ ( C = 2 and there are two ( ⊕ 2 . Then rank S e | ⊕ 6 T U utomorphism of order 24 2 E W . mirror Berglund-Hübsch − ), is given as follows (by ). ⊕ A 3 z W /J the surface σ W ( U ( ⊕ the action is given by (see the ) = while ) = 8 or 16. The second case W 3 8 T SL =1 ) = 14. ) is given as follows. On 6 ⊕ ), is also its own mirror in the X and the σ U 6 n E T 3 ( ( . Therefore, using either of the E 2 SL X = X | ⊕ σ ( Q S T on ( W A W, SL W ( 6 , S , G S J = 0 (2.34) T 3 E ) = = ⊕ W,J σ T 3 ! ! 8 or 7 ∼ = 24 = σ X on 2 1 1 E z ( ) = ) ) rank W T − 3 p ) and ( − − T 3 S ) and on 1 + J σ σ = rank σ W 2 1 1 0 W ( ( 8 1 ) = J = S S X − 3 y

2.31 ρ σ G

( – 14 – + . In this case W, J := 3 T 1] , 2 := 3 x )-polarized , A 3 3 8 ) admits a non-symplectic automorphism of order 3 st 2 , + σ ( M A 2 [12 ), the action of , either S = 1 hence P 2.34 w W W defined in ( | ), the action of is given in the appropriate basis over U W SL W ⊕ 7 W, J σ U /J 3 ⊆ , while keeping the other variables fixed. As was shown in [ x W M 3 G that gives (see Lemma ] that the W, SL SL ⊆ iπ/ z | 23 2 it is the same as for the previous surface, see equation ( ) is simply given by minus the identity matrix in twelve dimen e 24 is polarized by the invariant lattice W ), the invariant sublattice is U X J ( iπ/ 7→ W ), the invariant sublattice is W 2 ⊕ T e x W U Let us consider the hypersurface W,J : with on 7→ . ⊕ X 3 the action is unique up to conjugation by isometries). On the 2 W, SL 8 W, J U G z σ σ form a LP mirror pair. Indeed we have: E 1 ⊕ W U For ( For ( The surface defined by eq. ( For this surface It was shown in [ Likewise, the action of First, for the surface ( • • W,SL symmetry provides then the mirror pair ( egudHushmirrorBerglund-Hübsch and, polarized by choices of non-symplectic automorphisms of prime order, one finds that in the weighted projective space sense of LP mirror symmetry. Example 4. dental lattice is contained in the lattice On the other hand,acting the by surface also admits a non–symplectic a X for the mirror surface ( the surface acting as contradicts the previous inequality so that it is given by the matrix Remark action of companion matrix of the cyclotomic polynomial Φ while on taking the lattice metric (Gram matrix) one gets JHEP11(2017)084 T p W σ ] as th a 8 ng as (2.37) W,SL X -symplectic is well-defined 20 , 4 . This construction is into a non-geometrical lattice 20 . The previous theorem , T p 4 . 20 U σ , )-polarized surface 4 U while the automorphism sms of a surface and of its is of rank six and isometric T 2 ⊕ Γ . he lattice Γ ed in the next subsection, we ce Γ σ ⊕ 2 R ( and U )          4 ons that were obtained in [ σ p S          p 2 D σ ⊕ , these two surfaces, polarized by σ 1 1 ( − 1 4 ⊕ T − − 2 1 these vector spaces are orthogonal to D 8 ) admits an order two non-symplectic respectively, are also LP mirrors. The of sigma-model CFTs with K3 target − 1 E W ⊕ T p 2 0 0 and the 8 ∼ = σ 1 2 0 0 2 3 0 0 1 1 0 0 1 1 1 1 2 0 − ) E W − − − − − T 2 1 1 0 2 W, SL σ ∼ = 1 1 0 1 0 0 and − W,J ( ) − − p – 15 – S 2 1 0 0 0 X 2 σ 1 0 0 0 0 0 0 0 0 0 − σ ( 2 1 0 0 0 0 , associated with a given non-symplectic automor-          T 0 1 0 0 1 1 0 00 1 0 0 0 0 1 − X :=          6 . By the corollary E ], the invariant lattice of 3 R := ) 22 M T p st 6 ), hence σ E ( X mirrored automorphisms ( T T )-polarized surface ) admits also a non-symplectic automorphism of order 2, acti 2 ⊂ σ ( of the same order; by the theorem of the invariant lattice of the CFT defined by quantizing the non-linear sigma model wi has an action on the vector space . U S 2.34 -cyclic K3-surface p T 20 p p T 2 , . Following [ ⊕ p σ σ σ R 4 σ w 4 D of prime order. The BH mirror of this surface admits also a non ∼ = 7→ − p ) σ 2 w σ ( Inspired by physical considerations that will be illustrat The surface ( : S 2 -cyclic K3 surface target space. To prove that its action on t we will need the following proposition: p automorphism ˆ 3 Non-geometric automorphisms ofIn K3 this sigma-models section we define automorphism automorphism to automorphism will consider a mirrored automorphism that combines phism form a LP mirror pair. One can check that ( indicates that the acts on the vector space the invariant sublattices with respect to each other in Γ asymmetric orbifolds of Gepner models. 3.1 Mirrored automorphisms and isometries of the σ We consider a appendix) in a lattice basis in which the lattice metric is spaces, combining themirror, action and of study non-symplectic the automorphi corresponding isometries of the latti inspired by the non-geometric string theory compactificati JHEP11(2017)084 ] T ], p ⊕ ⋉ ⊕ M σ ) ) : 30 R T p ) (3.2) (3.3) 19 maps , L σ is the p , with (3.1a) (3.1b) 3 ]). ( ) . 1 σ ) p } ( T suggests − p isometry 31 σ 3 and T µ ( σ 13 p is called the ( , K T is unimodu- to the whole 7 S ∼ = , ⊕ . /M 20 5 T p , ) 20 ∨ , 4 , σ T 3 p R 4 Γ M , σ Γ is the determinant 2 ( has finite index [ and T L on the lattice ∈ { p M its LP mirror, regarded ) σ p ], Theorem 0.1 and [ T p det T 3 to its mirror; , ) can be defined. However 21 ,G he lattice , σ K L and if T p σ 3.1 W L generates the order ( ot yet complete (see [ . The quotient p X . σ M of the lattice ) , , that we associate with the action acts as p 7 20 20 , p , σ 4 ⋆ 4 σ ) mirrored automorphism ◦ on the vector space Γ is not in the geometric group O(Γ (Γ T p µ ⋆ p p σ p O σ ◦ ˆ σ σ -polarized K3 surface, where ), such that both lattices have the same rank. The ) prime, and T p act trivially on the discriminant groups of = = ( p L σ p is a sub-lattice of i ⊂ , its LP mirror, with σ R R ⋆ ) ( p T ) ) p ◦ is a diffeomporphism of the original p ˆ T T M σ p σ S p – 16 – 1 σ h σ , where for a lattice p ( σ | − ( if σ ( ) T T µ

T S p ⋆ M p the diagonal action by ⋆ and p σ -module that contains ) := ˆ σ induced by ˆ ˆ σ := ( , that we name 1 p allow us to define ‘mirrored automorphisms’ of CFTs Z -cyclic K3 surface polarized by the invariant sublattice p p -cyclic σ p p T σ 20 ˆ σ 1 p , ˆ σ action, with 4 O(ˆ p det | σ be a be a ) is a free = ,  | the lattice isometry induced by ˆ M ) , including the non-prime cases, the physical construction , polarized by 1 p W,G ) p is an over-lattice of finite index σ W,G T X ( so that we can extend the diagonal action by 20 X ,  T , and Proposition ( ) 4 T p Let 1 Γ σ ,G and Proposition det ( T Let | as a sub-group of the Abelian group of a lattice 1 T W is an over-lattice of a lattice 3 back to the original one, ∨ -polarized K3 surface. M X . L ) ], Corollary 2.6). By properties of K3 surfaces (see [ K ( M T p 20 , 30 σ denotes the BH/LP mirror involution which maps the 4 ( , is then given by and of Γ µ can be extended to an isometry of the lattice and -cyclic K3 surfaces target spaces in the following way: R S and so is non-geometrical. The automorphism ˆ ) ) p ) ) Due to Proposition By Corollary Corollary Observe that For other values of The equality of the determinants follows from the fact that t The action of the mirrored automorphism The isometry of the lattice Γ (viewing T T p p p p A lattice 19 7 σ σ σ σ , L ( ( ( ( 3 is a diffeomorphism of its mirror. the mirror dual lattice discriminant group of the lattice. in S lar (see [ lattice T the mathematical classification of these automorphisms is n with Lemma 2.5) the automorphisms index given by where as an of a CFT automorphism denoted Definition 3. Z of the lattice metric. T T Proposition 1. subgroup invariant sublattice under the that similar non-geometric automorphisms acting as in eq. ( JHEP11(2017)084 ; , . . ) v i ∈ 3 Z 2 T p on )– = 0 2 2.3 ∈ σ p ) p 2 h− ( (3.4) is an σ n ) and f σ ( T /k p = 0 ) 20 = T σ , b ⊕ ( 2 4 2 ) e w + T , resp. p i a σ 2 ( ) is therefore h ) ) and . To see this one ction of p T , hence on the acts exchanging T p U ι , ρ 20 σ 2.26 , ( 4 . One takes now the such that has been described in the discriminant T ), i.e. the restriction U f O(2 is straightforward to T p , 19 , e σ × 3 on Γ a/p ( isometries can be deter- ) p ). The lattice Γ T ) and the transcendental p , σ ρ T p 20 ⊕ , X mal compact subgroup 2, which has square σ ) 4 ( / by sending the two generators to with − ( U p ) e. This is in fact a more general Q on which we have an isometry ]. The corresponding matrices b σ T U ⊕ S ( 20 /p + 20 32 , ) , T U a b 4 on . It has a diagonal action on ) already discussed in example 3 into has index two in ⊕ + T with generators p σ ), and we ask also that (( O(2 i . i := ( σ 8 ) on 2 T p a Z 2 U 2.26 ⊂ T σ p w E ( id). The discriminant group of and are characterized respectively by a )) angles; these angles will be discussed one should add to the generators of the ice 2 and the induced involution ˆ , σ ) T T − p 1 on the K3 lattice Γ / p p , ⊕ ) σ i ⊕ h− 20 i ⊕ h− ρ σ ) – or equivalently into , 3 b ( 2 2 4 admits an orthogonal decomposition into the h σ T p h T − O( σ U 20 a ( := (id , – 17 – 4 × ⊕ ι Q a non-negative integer, of the sum 8 S U 2 = ( k ). / ) on these classes is then obtained by linearity (over ], explicit forms of the Γ , T p ⊕ | O(2) T )) p ) b 8 23 p , σ , σ -elementary lattices i.e. the discriminant groups are sums p , E + ) and σ × p 2 has determinant 1 and it is in fact isometric to ( p p σ classes of the form ( -cyclic K3 surface, all the relevant sublattices a / σ ( 22 σ p T ) ) ( k b p ∼ = − which acts as Q ρ + a i ) are on the transcendental lattice that was studied in subsectio det S 20 2 , a according to Lemma | − ). The lattice Γ T p 4 3 := ( σ σ Γ 2. One considers now the class C ( , in this way the lattice := v in the discriminant group b/ and ( 2.33 f T i ⊕ h− k O(20 ) exactly 2 a p − h and T p b/p ), ( e × σ on the sigma-model CFT associated with the surface ( w ( )) angles and a set of rank( ) and := 3 p T 2, resp. p 2.31 3 target space, is obtained by considering the geometrical a σ σ We consider the self-mirror surface ( b σ ) and a/ O(2) ( ( ⊕ as before). Then to construct Γ K which is induced by the isometry ( p ) T ) of this surface. T σ p k p ( and , X σ T ( ( k f ⊕ T T = 1. One can primitively embed the lattice ) + to that lattice is equal to ( ) have been tabulated in [ ; in particular it cannot be put in a diagonal form over ) and, for the BH mirror surface, the action of e Z Given that, for any given As the surface and its mirror are isomorphic to each other, it Explicitly, the action of the non-geometric automorphism ˆ T p ef p p w To make the situation more clear we consider a concrete exampl of order 8 σ σ σ /p := ( ( . The action of the isometry ( p Z ˆ the lattice generated by and takes the generators T is generated by CFT with a and over-lattice of index duplicating the action of T mined from lattice theory for any given example, see e.g. [ The action of ˆ The action of the geometrical automorphism can be diagonalized on set of rank( there, see eqs. ( Example 5. further in the following sections. a situation in lattice theory. We consider the hyperbolic latt invariant quantum lattices of ˆ (recall that define the action of the mirrored CFT automorphism ˆ isometric involution on Being of finite order, it is conjugate to a subgroup of the maxi lattice ( Z group of lattice the rationals). We get in this way a set of generators of Γ corresponding respectively to the quantum Picard lattice σ JHEP11(2017)084 . 8 ⋆ 8 ) E T E 3 σ (3.7) (3.5) (3.9) (3.6) (3.8) ⊕ 2 which is A 8 and ( ⊕ E ⋆ 3 U σ ⊕ , acting on the U . W ⊕ j ) ] with Fermat type 6 ) and its mirror we T p E , . σ 33 ( ) R , ⊕ ) of sigma-model CFTs 2.34 T 20 are given respectively in present context as some , U 11 8 4 8 2.1 ⊕ ms defined in the previous h isometry on ⊕ E ( 24 integer matrix E ) 3 ld [ p U ⊕ symmetry × O(Γ M σ : ) and more precisely the lattice 2 ( , ∼ = 1 ∈ 20 K 4 , A T ). ) k R 4 4 Z T p and      ⊕ Z O ℓ σ 2 U ( 2.31 U Z + A ∈ 3 ⊕ ℓ T 3 0 U ⊕ 3 k 4) superconformal field theory obtained 3 M ⊕        , ) , Z iπ/k 8 M U p 2 U E 3 e 8 σ 0 + ) to give an isometry of order three on ⊕ by eq. ( ( E and act trivially on the discriminant groups. 3 ) diagonal 3 2 on the sigma-model CFT associated with this M ⊕ ), as for the previous example. 0 0 U 3 3 T k 2 7→ U 6 3 2 R M ℓ Z ⊕ (see Proposition M – 18 – σ E 3.5 A ] is a (4 3 0 0 U Z , 6 3 U . Now the isometries of order three + 20 6 25 , 8 ⊕ M E : 1 4 M E 0 0 0 3 , . . . , k U k 3 E 1 1 U ⊕ M Z k W ⊕ M j in eq. ( , 0 0 0 U U 3 8 24 integer matrix: = U 3 ) and E ⊕ 3 M ⊕ 0 0 0 0 0 0 ˆ ). × M W U 3 6 M U = gcd( E 3 2.33 , respectively 0 0 0 0 M      2.37 2 M ∼ = K A := ), ( U 20 ). So the action of ˆ , 3 ⊕ R 4 ˆ 0 00 0 0 0 0 0 0 0 U 3 Γ M 2.35 2.33 M is then induced from the block-diagonal 24 ), (        3 By considering the K3 surface given by equation ( σ 2.33 is given by eq. ( 8 ), ( is a sublattice of index 3 of E 3 2 M A 2.31 A Gepner model for a K3 surface [ This matrix is an isometry of the lattice ⊕ 6 where the various matrices eqs. ( where chiral superfields as without fixed vectors, but up to isometry there is only one suc as the infrared fixed point of a (2,2) Landau-Ginzburg orbifo given by equation ( superpotential on K3 surfaces.of These them Gepner are points fixed play undersubsection. the a action special of role the in mirrored automorphis the 3.2 Symmetries ofThe Landau-Ginzburg Gepner mirror models pairs arise at special points in the moduli space E Example 6. have the orthogonal decomposition of the vector space Γ The action of ˆ surface is given by the matrix quotiented by the order The latter is a sublattice of index 3 of Γ given by the block-diagonal 24 They can be then combined (see Definition have no fixed vectors on JHEP11(2017)084 } 1 ] for − 8 tifica- (3.10) (3.11) (3.12) (3.13) model. -twisted γ ,...,K keeping the 0 ) by the sym- ) has a quantum ∈ { 3.8 γ etry group of the btained from this 3.8 ) contain states with del ( by nzburg orbifold of su- ld break all space-time , upercharges are charged ntial ( 3.11 orbifold theory (see [ flows to a super-coset CFT discrete torsion ll refer to this orbifold as the ℓ k mod 1 r in the spectrum. 6= 0. orbifold projection by assigning X b p p γ . = . Z γ 1 + φ . Z p W γ Q ]. iπ 42 2 iπ/p K/p 2 ) ≡ e 11 e q is a prime number. The theory admits the , σ 7→ = 2 minimal model CFTs, as every single- 1 7→ 10 k – 19 – 1 γ := ( N mod 1 and the non-geometric orbifolds of Gepner models φ Z := 1) supersymmetry, or, compactifying further on a q p : : , -charges whenever σ orbifold by the symmetry ( p q p 3.1 R p q p Q σ σ Z = (1 + p N , generated by: Q p We will now explain the relation between the mirrored auto- Z -twisted sectors. ≡ γ p ˆ Q ] which is not present in the large-volume limit of the sigma- 34 orbifold theory, the quantum symmetry acts on a field in the symmetry hence are projected out of the spectrum, ]. These Gepner models lead to IIA superstring theory compac p K ], following earlier works [ ) and assume that 35 = 4 supersymmetry in four-dimensions. [ Z 9 Z , ℓ 3.8 N 8 k orbifold. The twisted sectors of this orbifold are labelled ) that we described above by adding a specific -twisted sectors of the ), isomorphic to K b Z U(1) symmetry 3.11 / 3.8 ℓ p k non-integer left and right U(1) under the all the worldsheet operators corresponding to space-time s the At the infrared fixed point, the superconformal field theory o We consider the Gepner model corresponding to the Landau-Gi One can then modify the orbifold of the LG orbifold/Gepner mo Starting from the Gepner model, one defines the The Landau-Ginzburg orbifold/Gepner model with superpote • • model is anfield orbifold Landau-Ginzburg model of with a superpotential product of sector of the model as: Quotienting the Gepner modelsupersymmetry. by Indeed one this can automorphism see alone that wou in the corresponding order SU(2) two-torus, to Asymmetric orbifolds. morphisms introduced in subsection metry ( details): presented in [ perpotential ( tions in six dimensions with Abelian symmetry [ In the diagonal space-time supercharges coming from the left-moving secto diagonal and realizing the projection onto integral R-charges. We wi model ( One observes that one can define a subgroup of the quantum symm and will be referred to as to every state in the theory a charge JHEP11(2017)084 q p of or , a Q 2 T T p p with − σ σ (3.14) ], they p orbifold 9 Q p ce, to the Z persymmetry. del in this case. . This discrete Z orbifold. In those is the charge under as considered in the ∈ p q p p Z p Q ˆ Q n, that the by non-symplectic auto- der anges the geometrical au- worldsheet is preserved by automorphisms introduced and he Landau-Ginzburg model t. As discussed in [ p rsion. This construction leads ave used the charge σ vers is invariant. Furthermore mplectic automorphism ace-time supersymmetry from either with the action of = 2 minimal model are mapped es come from the right-movers. = 2 supersymmetry in four di- compactification is actually iso- p σ . N do not belong to the moduli space N -cyclic K3 surface (or more generically 6= 0 of the new p mod 1 b . In view of the discussion in section q p keeps only states with integer left R-charge. σ b p ). Similar constructions exist for Calabi-Yau K 2 ˆ − Q T – 20 – K ) are related to each other by modular invariance. Q ≡ 3.14 We have described in the previous subsection orbifolds K 6= 0 contain right-moving operators that, despite having ˆ keeps all space-time supercharges from the left-movers, b Q p ˆ ) and ( Q that we have noticed above corresponds, in the language of , and by projecting onto states with p q subgroup of the quantum symmetry of a Landau-Ginzburg orb- q 3 = 1 in four dimensions. 3.13 orbifold projection is modified, as one projects onto states Q σ p ± N K p Z -charge, have the properties of generators of space-time su Q correspond precisely, at the Gepner points in the moduli spa R projection w.r.t. the charge 2) superconformal field theory to the original Calabi-Yau mo := with its quantum counterpart , K 3.1 p p , to the possibility of pairing the action of of an order ˆ σ Z Q q p 3.1 σ is the charge of the given state under the action of p , where Q Z -cyclic CY manifold) by a non-symplectic automorphism of or However, in the specific case of an orbifold of a Under mirror symmetry, the right R-charges in every One can check, by inspecting the one-loop partition functio The two-fold choice of discrete torsion in the definition of t ∈ p K ˆ present work, the twisted sectors non-integral right Q Interestingly, the worldsheet model formorphic the as a non-geometric (2 of a sectors the diagonal torsion has also an effect in the twisted sectors the quantum symmetry three-folds, leading to ifold superconformal field theory is identified with a non-sy instead, in which case the invariant space-time supercharg tomorphism to minus themselves. As a consequence, mirror symmetry exch the corresponding BH mirrorin K3 surface. subsection Hence, the mirrored morphisms of the corresponding K3generically surfaces to with non-geometric discrete compactifications, to i.e.of that compactifications on smoothmensions (after manifolds, further preserving compactification on belong to a more general family of quotients of Gepner models generator where Hence space-time supersymmetry fromthis the orbifold left-movers with on discrete the torsion. Notice that one could h The charge assignments ( while none of the space-timethe supercharges diagonal from the right-mo subsection Fractional mirror symmetry. of Gepner models withthe discrete left-movers, torsion, and that preserve none all from sp the right-movers at first sigh of its inverse. projection w.r.t. the charge symmetries orbifolds with discrete torsion described here. JHEP11(2017)084 . 3 all for K 3.2 n the (3.16) 3.2 (3.15a) (3.15b) = 3 su- c -geometric of the same s orbifold, no ], generalizing fractional mir- 9 superconformal . We consider a 2 2 Y T , × 1 bed in subsection tor are indeed massive Y 3 es, called K pactifications of type IIA mmetry extends to a map ombining this type of ill have a four-dimensional 1 2 discrete torsion etric freely-acting orbifolds ory with the free for the reasons given in the nates for the geometry and π/k π/k . g twisted sectors. If one chooses ntegral right-moving ones, hence 2 s case the accidental isomorphism 3.1 + 2 + 2 ] was proposed in [ 1 2 2 36 Y Y R 1 2 k 7→ 7→ orbifold of this 1 2 2 k + ,Y ,Y Z 2 1 2 Z Z × , the masses squared of the two massive gravitini 1 – 21 – 1 1 2 1 k R 1 R 1 ]: Z iπ/k iπ/k k 8 = 4 space-time supersymmetry do not play a role, 2 2 e e and N = 1 7→ 7→ 2 R 6= 0 will only contain massive states from the space-time 1 2 Z Z m b = 2 space-time supersymmetry, all space-time supercharges : : ) above. As described there the discrete torsion is such that 1 2 N g g 3.14 ], between geometric and non-geometric compactifications i 9 ), ( orbifold theory. Because of the freely-acting nature of thi in [ 3.13 2 ]. k 2) non-linear sigma-model on a Calabi-Yau manifold and a non Z , = 4 supersymmetry are [ 37 × N 1 k -twisted sectors with b Z ] that we will now summarize briefly. The two gravitini obtained from the right-moving Ramond sec The starting point is a Gepner model for a K3 surface as descri We add to each of these two freely-acting orbifold actions a This isomorphism implies the existence of quantum symmetri In the following, we will focus on freely-acting orbifolds c 8 worldsheet model. A linear-sigma model description [ type as in eqs. ( perconformal theory with a two-torus target space of coordi model generated by a type IIA superstringMinkowski theory built vacuum upon with one ofbeing these obtained models from w the left-moving Ramond ground states. these models have integral left-moving R-charges but non-i of [ Consider the tensor product of this superconformal field the the field theorysuperstring are theory, the whose building vacua blocks correspond of to non-geometric the com non-geom Landau-Ginzburg regime. Outside ofbetween the a Gepner (2 point such sy the ideas of [ As it is, thisprevious subsection. orbifold breaks all space-time supersymmetry ror symmetries freely-acting supersymmetry-breaking massless states could possibly arisean orthogonal from two-torus the of correspondin radii in the of broken non-geometrical orbifold with a shiftand along a corresponding circle; restoration in thi of as the 3.3 Worldsheet construction ofThe non-geometric mirrored backgrounds: automorphisms summary described in subsections point of view. JHEP11(2017)084 . 2 p × k ns, from 20) , U(1) ; in the / 5.2 2 2 li are the k 3 SCFT, con- = 2. We will O(20) subgroup K = 4 supergravity the moduli space × 1) supergravity in N ss hypermultiplets. , N and SU(2) 1 3 to 6 dimensions and O(4) k It was found there that = (1 K , that are part of space- 9 ), i.e. the isometry group S supergravity model at low ]. N e these constructions from on from Landau-Ginzburg 20 9 U(1) , / 4 rings of the paces in subsection tion that corresponds to the tensively in the supergravity 1 ils elsewhere. emaining constructions, some mpty and hence all the moduli k a duality symmetry O(4 ecial cases, and will give some ) STU in a gauged s in moduli space by a compact- dered, for prime and non-prime order We have considered type IIA su- 3 gives K ]. 5 ). 20) monodromy round each circle gives , Z 20; , = 4 supergravity scalar potential are possible. identified under certain automorphisms. This 3 surface as was discussed in section D – 22 – 2 K T with an O(4 × = 2 four-dimensional 2 3 3 moduli space which is a fixed point under the au- N T K K with duality twists with non-geometric monodromy. In the 2 ]. We will here discuss the supergravity limit of this, which 2 T ] and references therein. For our string theory constructio = 4 supersymmetry in four dimensions to 40 – N 38 , 5 3 SCFT have become massive; the only remaining massless modu K moduli of the two-torus and the axio-dilaton modulus 3 moduli survive and appear in the low energy theory in massle = 4 gauged supergravity from duality twists 20) as it is for these compact monodromies that fixed points in ] all pair of purely non-symplectic automorphisms were consi U , K 9 In about half of the possible constructions, this subset is e We will focus here on the case with monodromies that are in the We consider then type IIA superstring theory compactified on The massless spectra of all these models were computed in [ N In [ 9 and . Then further compactifying on minimal models; we will consider the corresponding moduli s taining states built out of the identity operator in the SU(2 the monodromies are required to be in the dualityphysics group literature O(Γ it is often refered to as O(4 is a dimensional reduction of Scherk-Schwarz type [ the massless states are identified with a subset of the chiral another perspective. of the lattice of total cohomology of the of the original ification with duality twists [ tomorphisms. (These fixedorbifolds.) points This were construction found is in extended the to last general point secti time vector multiplets. It gives the requires being at a point in the then further compactified on the low-energy four-dimensional viewpoint. 4 energies (compared to the inverseof size the of the torus).We now In turn the to r the second part of this article, where we analys corresponding to Minkowski minima of the As we shall see,vacua some interesting that features break arise the for these sp of O(4 R In this section westringy study construction the considered in supergravity theperstring dimensional previous theory reduc sections. compactified on summarize the supergravity results here, and give more deta supergravity limit, compactifying IIA supergravity on six dimensions coupled to 20 vector multiplets, and this has T in four dimensions. Thisliterature; construction see has e.g. been [ discussed ex a Scherk-Schwarz reduction of the supergravity, resulting JHEP11(2017)084 . ai I m × G λ ent 22) A itini , ∈ . The O(6) / which is H 6 of these H, g 22) global V , 22) t . From the O(22) while , ∈ g nsformations V , which takes × h V τ O(6 1 = 22) must be the × − , , four gaugini h a M O(6) 22), such that they A / , to be a lattice metric . The group O(6 . The scalars consist s g 22) V 7→ , giving rise to a gauged H O(6 ctor , = SL(2) g to the 28 vector fields MN M roup) can be obtained by with monodromies in the G t η g G V ∈ ed case. Simple dimensional and can be represented by a om the discrete monodromies a local symmetry which in the take has ce O(6 O(20). The 24 vector fields O(22) as M 7→ 20) (and a further rigid symmetry vity, but we will restrict ourselves to this under the action of × , G/H : × U(1). The scalars in O(6 1 and a complex scalar G / ). However, in the next section when − (4) 20) and are invariant under i = 4 supergravity coupled to 22 vector h , 22 χ with twists in I ˆ V = O(4 2 1) supergravity coupled to 20 vector mul- g 22) symmetry (of dimension 28 at most) − P in N , , , T G 6 = I O(22), acting on the bosonic sector through = O(4 s ˆ on the coset space given by V → = (1 × – 23 – H G = O(6 20) but transform under 22) and for the supergravity theory we can choose N M , , (6) of G P in 24 gives rise to 22) and local O(6) , 2 2 = O(4 T T O(20). In extending to the fermionic sector, the local sym- G × transformations. = O(6 transforming as H of the rigid , four spin-half fermions = 4 supergravity multiplet contains the vielbein, four grav of signature (6 G For this to work, the vector representation of O(6 ˆ . V 20) and in the following section we will consider the consist m µ 3 , K A N 10 ). = O(4) MN d and transforms tensorially under η + H . H SO(2). The SL(2) acts as usual through fractional linear tra U represents dim(G) = 276+1 degrees of freedom, but dim(H) = 19 and scalars taking values in the coset cτ / ( ˆ V ⊕ φ / 24) transform as the ) b U , one can construct a metric + V ⊕ O(22). The ,..., 20 aτ , × ( 4 A gauged version of this supergravity (with electric gauge g The 22 vector multiplets in four dimensions each contain a ve We now turn to dimensional reduction on In this section we will consider the supergravity reduction Note that this is not the most general gauging of the supergra = 1 , six graviphotons 7→ 10 -valued vielbein field i µ I choosing a subgroup and promoting it toalready in a the local theory. symmetry, using a minimal couplin on Γ we apply the supergravity analysis to string theory, we will invariant under vielbein a basis in which this is the diagonal metric ( transform under global preserves a metric the remaining two parameterize the coset space SL(2) τ O(22) can be conveniently expressed in terms of a vielbein and six real scalars. 132 scalars parameterize the coset spa value in SL(2) can be removed by local multiplets in four dimensions. The massless Abelian theory of a dilaton G The vielbein supergravity in four dimensions.reduction (with Consider no first twists) the on untwist fermions are invariant under ( consisting of constant shiftsbosonic of sector the is dilaton). There is also ψ O(6) symmetry, and a local symmetry metry is actually a double cover of this, tiplets, and this has a rigid duality symmetry The starting point in six dimensions is type IIA superstring theoryconstructed in compactifications section that arise fr continuous group O(4 4.1 Twisted reduction on class of gaugings here. JHEP11(2017)084 = to 22) 2 al is , (4.4) (4.3) (4.2) (4.1) (4.5) 4 are T is the trans- under a M,N G ( ,... city, with ) under a MQ ψ MNP ] x t ( K . g M [ 0 1 = 1 ψ nd the tensors R /R transformations. MNP h.c. 2 i, j t 7→ ) + G iR τ h.c. a j ψ -dependent ( . Then the complex upergravity on preserves the O(6 3, are the coordinates 1 2 λ y = , ℑ + U(1) and µ ed in the introduction. b j G / K Γ U 24 = 4 field πR . Here λ i µ t a i ¯ ,..., ij + , ψ ng restrictions on the sub- ¯ D λ through j ) ab + 2 . ij x ai y 2 the addition of a scalar poten- 2 = 0 ( 2 ab PS ,A y y 0 , j 2 of the gauge group transformations are required to A µ L A ai ) ψ N 2 , 1 ∼ P − e + )] µ y G NR 2 ( j 2 x ,A 1 L y Q aj y χ ij 2 ( g MN λ ) = µ ) t 2 2 a i 2 g Γ NP MQ and ,A [ y ¯ λ y t ( µi and ( ij 1 2 ij 2 R 2 ¯ 2 M ψ 1 A T g A 3 )] MQ ij 2 , the scalar in SL(2) 1 2 1 η τ y A − πR , g ( 1 3 ) = = + – 24 – 1 1 2 y j g PS [ y 1 ) + + 2 ( a N R 2 1 M e λ νj ) around the first cycle and a g y MNP ( ) 1 i ψ t 22). The gauging of the four-dimensional supergravity 1 y ¯ NR , χ ) = ∼ ( y i µν ) = j ( 1 1 ) as the transformation of a Γ 1 1 ai M g i , y y 2 g and complex structure modulus is y O(6 ( µi µ ] and references therein. In particular, the scalar potenti ) transforms in a representation of , y 1 A ¯ 2 x ψ i . Then the Scherk-Schwarz ansatz is MQ g µ ⊂ ( 43 − R ij 1 R are coordinates on O(22) discussed above. x , y – ψ 1 ( M i A K µ = H × ψ y 41 1 3 x iR 2 ( ∈ 2, with / which is the condition that 1 , ψ = = h QRS L 2 = 4 supergravity. We consider a rectangular torus for simpli O(6) t 1 T / ; see [ / 3 MN − = 1 ij N η e L ) around the second cycle. The two transformation i ab 1 ) (where 22) 2 i MNP , = 6 field , − y i t G e ( ,A ) y , y 2 j D µ ai τ g ) denotes the imaginary part of that can be gauged, and fixes the form of the scalar potential a 2 ( x -dependence is taken to be exponential, so that τ 1 ( ℑ ( y K ,A ψ ℑ 48 ij 2 , together with fermion mass terms given by 28) which satisfy the Jacobi identity and the constraint tha We introduce twists around each of the two circles as describ We now turn to the twisted reduction of the six-dimensional s V = ,A ij 1 -dependent ,... V is completely antisymmetric. Supersymmetrytial then requires ¨he modulusKähler is coordinates invariant metric obtain a gauged giving the 1 rigid transformation of the four-dimensional space-time) is taken to depend on is completely specified by the structure constants A field where metric on O(6 Specifically, suppose in terms of certain scalar-dependent tensors SU(4) indices. Supersymmetrygroups and gauge invariance put stro y adjoint representation of A and formation commute, and the JHEP11(2017)084 s as H and and and (4.8) (4.6) (4.9) (4.7) M 1 O(20) T A 12 N × dependence, 20) invariant O(4) , 2 O(20) so both y / N × 2 22) vector 20) , ], and here we will . This picture de- , πR ] = 0. In the six- 2 2 G e 44 ,N ], the fixed point gives a g monodromies to bring the fixed point . 1 transformation must be 2 ]. The O(4 s ) = . 1 N 2 nder i f [ n in [ H are the vector fields − G 44 ) = 1 is O(4) ,[ X -vanishing commutators are 0 , πR G dependence, and this requires ˆ 0 G V K s, giving a Minkowski vacuum. to be the metric on the lattice ˆ IJi (2 V y er iI 2 n of [ N f g = ( IJ 1 η , g − JK symmetry is not fixed. Choosing a ] = P L J T . H (0)) P = ,T 2 I g    ( T i [ i in the moduli space O(4 MN I . These then get non-trivial IJi t Z ˆ T X V 0 can be combined into a O(6 ˆ 20) preserving V ,    ). However, in the next section when we apply ] = , – 25 – K , = 1 N 20 = J N I ˆ ,N V T 1 ,T J − of the Lie algebra of M J , to one in the double cover iI T M πR iI 4 2 transformation that act on the fermions through an 2 I T N [ e N H ,N 24; for the supergravity theory we can choose a basis in H ] = symmetry would mean that a 1 O(20). These are then in a Cartan subalgebra SO(2) ] = J ) = 44 20) transformation with N I 1 × , H ,... iI t ,T i πR = 1 Z [ (2 1 g is then [ 1 I,J − K ) must be in this subgroup. = 1. The subgroup of O(4 (0)) 0 4.6 1 ˆ where V g ( IJ η . The generators of the gauge group Full details of the reduction for the bosonic sector are give The supergravity reduction is then specified by two commutin Suppose now that there is a point in the Lie algebra of O(4) 20 , 2 4 to the origin, the scalar fields, represented by the vielbein transformation, so that in this gauge the fermions also get metric is the choice of a lift of the twist in dimensional supergravity, the only fields transforming und monodromies ( Then we can perform an O(4 minimum of the scalar potential where the potential vanishe just quote the results needed, mostly following the notatio and all other structure constants are zero. Then the only non for two commuting elements which this is the diagonal metric ( Then the monodromies are where the structure constants of the gauge group are N pends on using thephysical formalism gauge for in the whichaccompanied local the by a local compensating the supergravity analysis to string theory, we will take while the fermions and graviton do not, as they are singlets u that is fixed under both the monodromies. From the arguments o Γ The Lie algebra of JHEP11(2017)084 ) ) ⊂ 24 24 will , , 12 (4.14) (4.13) (4.12) (4.11) (4.10) 1 1 m ( . i ) + ( q O(20) and has charge 1 O(20) these i , × A 4 × . , , . , the mass . 12 ! ! O(4) ! ! ) rged under U(1) 12 12 12 12 , where 12 12 0 0 × ˜ i ˜ θ θ θ θ ˜ θ θ 20 A , 4 12 12 sin sin cos cos ing the charges . ˜ , will remain massless. 0 0 θ θ O(2) 1 ). 2 i − − 12 12 , ˜ 12 12 θ θ A ! . × . 1

˜ θ θ 2 i 2 ˜ θ sin sin specify monodromies in the O(4) i ! cos cos πR q 12) under U(1) ) + ( 2 − − 2 2 i 1 ˜ θ ˜ θ 22) O(22)

, i , , ⊗ · · · ⊗ ⊗ · · · ⊗ πR q 1 4 =1 12 ,... × i 2 ˜ X θ , given by ! ! 2

2 2 O(6 , 0 0 ˜ = 1 θ θ m 1 + + and O(4) 2 2 22) and this decomposes into ( ⊗ · · · ⊗ ⊗ · · · ⊗ 2 2 O(22). ˜ , θ θ 0 0 2 i, j ) vectors massless. The 24 vector ﬁelds in the ! ( ⊂ ! ! − − ⊂ , then the vector ) + ( 1 – 26 – , θ 1 1 1 1 i 1 1 i 22). Under O(2) i i , ˜

1 ˜ θ θ θ θ q , θ θ θ 4 20 i i πR πR ⊗ ⊗ , q q of O(6 2 2 sin sin O(2) 2) has mass cos cos 1 O(20) ) representation are singlets under O(4) O(6 , ! ! , O(22) can be parameterized by ﬁelds transforming as i 1 1 1 1 1 × × 1 28 0 0 ˜ 1 1 =1 , ˜ 12 θ θ θ θ A i , O(20) conjugation) in which they are both diagonal and twist. These scalars parameterize × ⊂ X ˜ θ θ = 1 O(2 2 1 1 ,

2 ˜ × θ θ sin sin 12 0 0 2 cos cos O(2) , m − − = − − O(20) are then 2 O(6) ) + ( 2

O(22) × 1 . Then m i , = = = = 1 × 22) 1 2 2 1 , 6= , O(6) and O(2) N N N N 2 j 1 2 , 2 1 20). A twist with all angles non-zero makes the vectors in the , 2 ( are both zero for some πR πR × ⊂ πR πR 2 2 i 2 2 e e ˜ O(4 θ , O(4) i × = 0 for 11 θ × j 2) q , ) under O(6) O(20). For a state with charges 22 , ) representation can be written as 12 complex vector fields × 6 The scalars in O(6 The 28 vector fields are in the These twists will in general give masses to fields that are cha Here O(2) 24 = 1 and , 11 i 1 If the angles we can choose a basis (by O(4) the ( Using this formula, the masses of all fields can be found by find be given by decompose as under O(2 O(4) representation massive and leaves the ( Of these, only those in the ( We choose a basis in which the angles factor and the remaining angles specify monodromies in O(20 q each specified by 12 angles: hence invariant under the U(1) The monodromies in O(4) ( JHEP11(2017)084 ; 3 = , M (4.16) (4.18) (4.15) (4.17) (4.19) amn U(1)] ). This ) where t / ⋆ A ) O(20) and , the mass , q kj has charges 4.12 ] 28 × M I n x to an anti- q G = NB ([ φ O(4) = ( V ik / i ] , some values of the ) into indices q m } is 20) G , 12 labeling the Cartan 12 [ µi 2 iven by eq. ( M,A gives direct access to the τ ψ ss bosonic fields consist of 1 . , 1 ,... √ vides the mass term for the ,..., 4 2 2 A 3 − ! ! = 3 = 1 12 = ( A A has charges = ˜ ˜ ). Then, for example, the 80 real θ θ 2 2 A , i j A where the scalar ]. i ). At the origin, ] − + U(1) ai 4 q ,... NB SL(2) πR πR , q 2 { ρ 2 2 45 4.2 h.c. M M kj 1 M MA ˜ ˜ θ θ q A ∼ = 1 + , ρ ), ( ik ,A i

1 νj = ( while 4.1 A ψ + + i MA q 2) O(2) φ 2 2 mnp µν = t , AB ⋆ 2) and 6 scalars in the coset space [SL(2) Γ × δ j – 27 – ) , i A A µi ) lj has mass squared θ θ 1 1 = O(2 ¯ ] 2 ψ p + − A ij 1 O(2) M πR πR G q ( MA A 2 2 ([ of O(2 M M φ × 1 3 U(1) are also uncharged and remain massless, so the θ θ kl 4 / ] and n G = = [ abm U(1) ⋆ t 2 2 SL(2) MN ) . Then δ ij m m ] ik ] m = AB ) representation take values in the coset O(4 m δ G [ G M 20 − 2 q , ([ τ 4 = 2 1 , τ √ A 1 1 2 √ q , 8 − 1 can lead to accidentally massless scalars. ) where = = and are the ’t Hooft matrices used to convert an SO(6) vector inde i A is a complex symmetric matrix. The mass matrix for ˜ θ ij ij 2 has mass squared , q , m ij 1 A i ab MN θ G M A δ A = q MA ρ All other scalars become generically massive, with masses g We now turn to the fermion mass terms ( It will be useful to decompose the indices Then for generic twists with all angles non-zero, the massle ij 1 = = ( 2 labeling the Cartan subalgebra of O(4) and indices A , M i fraction of supersymmetry preservedgravitini by given the by vacuum, as it pro formula indicates also that, for a given set of charges q 1 The axion and dilaton in SL(2) remain massless. where and matrices of the model simplify considerably to give [ scalars in scalars in the ( symmetric pair of Spin(6) = SU(4) indices. The first matrix where q angles subalgebra of O(20), so that the charges are can be written in terms of complex scalars the graviton, 4 vector fields in the this is precisely the bosonic sector of the STU model [ JHEP11(2017)084 s ). H 22) 20) ) = , , 2 4.12 (4.20) (4.21) 22) to , q ) while , 1 O(4) are q lculation is zero or ⊂ 4.20 2 O(20). The 2 m ⊂ 10 ) on using ( ). The remaining O(4) are ( . 4.21 given by ( 4.21 so that ) ⊂ 1 2 2 20 ), ( ˜ 2 fields transform as θ = 4 local supersymmetry m , 20) transformations must / , 2 = 2 supergravity multiplet sent case it does not occur = . , N 4.20 2 1 1 2 N = 6 supergravity has a local ˜ w-energy sector of the theory. θ ! transformations. The gravitini ! given by ( 2 the 2 2 ˜ symmetry and a global O(6 ctrum of the STU model. Unlike s , and if the O(4) monodromy is θ D 2 = 0 corresponds to a monodromy 2 ˜ θ ) + ( 2 , but do not transform under the s H and 2 m − s πR + O(20), and as a result they become, H 20 πR 1 2 m 4 H 1 , ˜ 4 θ θ 22) will break the global O(6 × ˜ θ 1 , 20) symmetry. The fermions transform ⊂ , SO(3)

) representation under U(1) =

, 2 1 + × 1 , O(6 + O(20), the spin-1 θ 2 1 2 SU(2) 2 , O(4) but not under the U(1) × ⊂ ) + ( 1 × O(20) 2 1 2 1 ⊂ θ , 1 are non-zero therefore all the fermions become , both with degeneracy two, where θ – 28 – × 2 2 symmetry is fixed, O(4 2 2 fields can be found by calculating the tensors − ) 2 , πR + s / 2 survives as a symmetry in the gauged supergravity. πR 1 4 1 SU(2) m 1 O(20) ) representation under U(1) 4 θ ( H 1 m θ transformations. As a result, reductions with O(4 1 ), or by finding the twists in the gauge-fixed theory. × , × × s SU(2) is zero. In either case, there are 2 massless gravitini and 1 and = H 4.2 , 1 = × 2 = 0 above. Similarly, and ( 2 1 ) representation will all get mass 2 ) m 1 ) 2 ) + 3 1 1 ) under SU(2) m 2 ) , 1 1 m 1 m , 1 , m ( , 1 ( 2 m = 4 theory, there is a local , 2 , , then SO(3) surviving as a symmetry. In the formalism with the local ( 2 2 1 so that ( ]) ( N 2 ) representation will all get mass × 2 1 × ˜ 45 θ , )+( . This is a Hermitian matrix whose eigenvalues are (after a ca 3 2). This then results in the mass formulæ ( − 2 ∗ 1 / , ) , 1 O(20) = SU(2) = 1 ij 1 1 ( O(20) symmetry and a global O(4 , − 1 A × , ˜ 2 θ × × 2 / 20) symmetry. If the local = ( with SO(3) , (4) and ij 1 1 2 2) while those of the gravitini in the ( θ A / ) = (1 P in = 2 supersymmetry, and the massless fields fit into the 1 − 2 2). However, O(4) has a subgroup SO(3) , For the ungauged For generic angles, both We see that something special happens if Similarly, the masses of the spin-1 These formulæ can be understood as follows. The , = 2 , q N = / 1 s 1 q ( (1 after gauge fixing, twisted under the U(1) transform as ( charges of the gravitini in the ( massive and all supersymmetry is broken. twists result in twists of the fermions by compensating in SO(3) six massless spin-half fields. In this case the vacuum breaks θ restricted to be in SO(3) This corresponds to the case O(2 to using formulæ from [ be accompanied by compensating symmetry. A monodromy in O(4) and fermions will all be massive for generic angles. Here we do the latter. Under SU(2) those in the 3 with three massless vector multiplets, whichmost is occurrences just of the spe theas STU a model truncation in of string a theory, richer in theory, the but pre describes the whole lo appearing in the mass formulæ ( where H The fermions in the 3 global O(4 under Spin(4) JHEP11(2017)084 . L 4 = 4 + (4.22) (4.23) R SO(2)) SO(20) D ¯ 4 survives × × 1 ∼ = 2 theory 2 ) 4 1 ) N = 0, then the , less states are survives as an 1 1 1 SU(4) , , 2 SU(2) SO(22), and the , i.e. 2 m 1 , ¯ 4 ⊂ ( × × 1 ( 1 × O(22) symmetry has × = 2 supersymmetry for × ) the R-symmetry for the 1 scale set by the gravitini ) + 4 N SO(20) ) ) SU(2) = 0 then the monodromies ) , O(20). If 1 l SU(4) corresponds to left-handed 1 1 ) + 2 1 2 2 , × SO(20) , , ⊂ , , 1 × 2 ) = 4 supergravity supermulti- 2 2 2 4 m , 1 2 , = 2 multiplet structure. The = 4 multiplets of the × 1 , , ts of , SO(20) 1 ) 2 , 2 1 1 , 1 N ( 1 1 ( × SO(20) and hence SU(2) 2 , N N , ( ) + ( × 2 1 1 × × SU(2) SU(4) 1 , ( SO(20) 2 ) + ( ) + ( × , 1 SU(2) SO(22) transformations. We now give 1 1 × ( ⊂ 1 × × , , , 1 2 × 2 ) + 2 × 1 1 1 SU(2) ) + 2 , , ( 1 SO(20) and hence SU(2) ) + 2 SU(2) 2 2 20 × 20 ( ( × , × × , 1 20 O(20) 1 2 1 theory , 1 2 , , × 2 1 SU(2) – 29 – , 1 ( 2 ( 1 = 2 SU(2) × SU(2) SU(2) ) N ⊂ 1 ) + ( ) is zero, and study aspects of the effective ) ) ) × SU(2) , 1 SO(22) 1 1 1 2 ) + 2 1 1 , 20 , , , × ( ′ , × m 1 4 6 1 4 20 1 ( ( ( ( × and right handed ones transforming in the , , 2 2 4 2 O(22) global symmetry, of which SU(4) is an R-symmetry. SO(20) , SU(2) ( 2 × × ( SU(4) 2 ⊂ 2 2 . 2 1 0 / / is zero or R 3 1 = 4 supergravity theory fit into an 4 1 ) ) ) m + D SO(22) 22 22 22 L SO(20) , , , = 4 vector supermultiplets. Once the local SU(4) × ¯ 4 4 6 1 × ( ( ( = 2 supersymmetry. ∼ N 1 survives as an R-symmetry in the low-energy theory. The mass ′ 1 4 = 0, then the monodromies lie in SU(2) N SU(4) 1 2 m 1 0 / 1 If = 4 supergravity multiplet is and SU(2) Similarly, decomposition of these representations into SU(2) the representations of the component fields under this globa N the cases in which supergravity decompose into massless and massive multiple The fields in the been fixed, all fields transform under rigid SU(4) valid at energies muchmasses. less than the supersymmetry breaking 4.2.1 Massless multiplets representations, which will be useful for studying the unbroken 4.2 Aspects of the low-energy R-symmetry in the low-energy theory. The surviving SU(2) is plet and 22 In this subsection we give further details of how the lie in SU(2) as an R-symmetry in the low-energy theory, and similarly if fermions transforming in the monodromies lie in SU(2) For the gravitini and spin-1/2 fields, the representation while for the 22 vector multiplets we have The monodromies are in SU(2) fixed, there is an SU(4) JHEP11(2017)084 of and 20 (4.26) (4.24) (4.25) (4.28) A θ − = M θ y the vector). lets. lves in massive multiplets. neutral under the monodromy ) = 2 vector multiplets, which is ) 20 ), a supergravity field with charges , N 20 1 , , ) ) ) ) ) 2 4.12 1 ) 1 1 1 1 1 . , ( ) ) ) ) , , , , , 1 1 1 1 1 1 1 1 2 2 = 4 ungauged theory are all in a × 20 , , , , , , , , , = 0 (4.27) , ) this will be the case if = 0 1 1 2 2 1 2 1 1 2 i 1 i , , , , ( ( ( ( ( N θ ) + ( , ˜ θ i 1 2 1 2 i 1 ( ( ( ( q × × × × × ) + 2 4.15 q ( SO(20), i.e. 20 – 30 – 4 3 6 3 2 , =1 12 20 × 1 =1 12 i X 2 2 2 2 , i X , 2 / / / / 2 1 0 1 0 2 2 3 1 3 1 , ( 2 SO(20) form a BPS hypermultiplet and states in the ( 2 × will be massless if 0 1 / 2 1 12 ) of SU(2) 20 , 2 = 2 supergravity with three massless 12) under U(1) N ,... . A ˜ θ = 1 − ) give a BPS massive vector multiplet (one scalar gets eaten b The remaining fields from the original = i, j 20 ( , M i 1 ˜ ( q States in the ( This gives a BPS gravitino multiplet and two BPS hypermultip In generic models,The the massive remaining states states that are now singlets organize under themse SO(20) are: This gives precisely the content of the STU model. 4.2.2 Massive multiplets and SO(20), namely: the ones that are singlets under SU(2) 4.2.3 Accidental massless multiplets In certain models, a fractionand of are the therefore BPS massless. hypermultiplets are From the mass formula ( For example, for the scalars with mass ( θ JHEP11(2017)084 ) 4.27 (4.32) (4.30) (4.33) (4.31) (4.34) (4.36) (4.35) (4.37) ]. For a 2 , each field 2 , y ) with charges 1 x y ectrum contains ( 2 . ,n with 1 2 n m ! φ ) (4.29) i formulæ. The gravitini ˜ θ x i sless states were ( ( q 2 ve, this formula is modified ,n coordinates luza-Klein modes [ 1 =1 2 12 i n φ P 2 πR . 2 + /R 2 2 π π 2 π π , π . y π π . 2 2 2 πn ) then has mass in n 2 R x + ( 1 O(20). For a mode

2 /R ,n × will be given by the following modiﬁcation + + 1 1 y n 2 2 1 m φ ! in = 0 mod 2 = 0 mod 2 = 0 mod 2 = 0 mod 2 = 0 mod 2 i = 0 mod 4 1 – 31 – O(4) = 0 mod 2 1 e θ i A A i 2 n 2 i R A A θ ˜ θ θ ⊂ θ q θ ˜ ˜ i θ θ ,n i q 1 q X + − + n 12 + − =1 1 12 i = 1 =1 12 =1 M M 12 i X θ M M 2 i X θ θ P ˜ ˜ , the condition that there be a massless KK mode is that θ θ ) = πR , the mass m 2 2 + . The mode 12 2 MA , y 1 1 φ , n πn 1 , y 2 µ n x

( φ = 2 m 12) under U(1) the condition is ,... MA ). Now we see that the condition that the full Kaluza-Klein sp ρ ): = 1 4.28 For a reduction with duality twists of the type discussed abo For the hypermultiplets For the gravitini, there is a similar modification of the mass i, j 4.12 ( i with a sum over integers and ( massless modes is that trivial reduction without monodromy on the two circles with and There can be further accidental massless multiplets from Ka 4.2.4 Further accidental massless multiplets from KK modes In the truncated supergravity theory, the condition for mas has a mode expansion of the form and and while for q for fields that are charged under U(1) of ( KK modes will include massless spin-3/2 fields if JHEP11(2017)084 1 ], n- 2 20) /p , 1 (5.1) ]. In given cyclic 2 )- πR 20 2 , , and we 4 , p 20 , 1 ying the IIA 4 p ), isomorphic eduction now for the second 20 2 , ] summarized in 4 8 ge lattice [ /p 2 πR gives the resulting low 4 o extend this to all points e last section to the non- nding to the ( bration of K3 surfaces over on t symmetry groups O(4 c type IIA compactifications, c twists, in the sense of [ ructions of [ nt in the moduli space of CFTs , from the algebraic geometry and ) ; the conditions for anti-periodic combined with a shift of 2 4 2 3 1 π ) associated with the one-cycles of , y , x π , π . p 1 3 20 σ ) preserving the lattice Γ , y x and 4 is the metric on the lattice Γ ( 20 , f 2 4 IJ + ) of the duality group O(Γ η 2 2 p p 2 σ x combined with a shift of 2 = 0 mod 4 – 32 – = 0 mod 4 = 0 mod 4 O(ˆ 2 + 2 generating an automorphism group isomorphic to p 2 2 θ 1 ˜ ˜ θ θ × σ p 2 1 p ) − x + − 1 σ K3 surface, i.e. a hypersurface in a weighted projective 1 p 1 1 θ = ˜ ˜ σ θ θ under the action of ˆ ). one can associate to them non-geometric automorphisms and W 2 3 1 cyclic 3.3 p T − σ ) × ), ( 2 3 , p 3.1 1 K we defined orbifold compactifications consisting of identif p are prime numbers. As we have seen, such surface admits two no 3.3 20) is broken to the group O(Γ 2 , p ; see eqs. ( . As a result, the mass parameters introduced in the twisted r 2 2 p and . Using Definition 22) are broken to the discrete subgroups preserving the char Z 2 . , generating a subgroup O(ˆ 1 p 3 p × Z 2 p 1 3 that is a fixed point under these transformations, correspo p ˆ The reduction with a duality twist construction gives a way t In subsection We start with a ( σ × with two non-geometric monodromies in O(Γ , K Z 1 1 2 p p ˆ in moduli space, and then the supergravity analysis of secti in section K3 surface at a Landau-Ginzburg point. for the first one-cycle of the torus, and ˆ one-cycle of the torus. Thison is all defined at a particular poi to choose the natural basis in which the metric the torus: and O(6 energy effective field theory.T The twisted reduction gives a fi superstring theory on section take discrete values. Our aimconsisting here of is to K3 find thethat fibrations non-geometri at over fixed two-tori points with of the non-geometri twist reproduce the orbifold const string theory viewpoints. In string theory,particular, the O(4 non-compac We will now applygeometric the compactifications supergravity analysed framework in developed sections in th ones would be slightly different. 5 Compactifications with non-geometric monodromies σ and and Z or These are for gravitini modes that are periodic in symplectic automorphisms space defined by a polynomial of the form where JHEP11(2017)084 24 ), by (5.2) (5.4) (5.3) × given on the 20 i has the 1 θ given by which in p ) at the − 8 σ i has eigen- 1 , θ E 3 4 N are given by U 2.26 of the mirror M ⊕ J T p U 3 iI σ t M 20) can be brought to , for the other circle from ld give a monodromy ma- ) matrices associated with , 2 O(4 Z , or equivalently can be writ-      upergravity. ⊂ πN ror K3 surface ( 2 C ). These give the monodromies , and this gives the structure U e 7 circle with monodromy ). The matrix ⊕ ) with the twelve angles is obtained using the formalism σ 0 3.1 1 U 3 y ) element is given by the 24 2 O(20) or M 2.31 20 4.11 T , ), associated with the non-geometric ), associated with the non-geometric 8 × 4 3 2 1 3 × E . 3 σ p p 0 0 ˆ , the GL(24; J σ σ 3 M 2 M (ˆ (ˆ ) the structure constants iI K O(4) = O O by eq. ( U ), see eqs. ( N 4.8 1 ⊕ ⊂ T p U 0 0 0 N = U 3 σ 1 1 ⊕ – 33 – ( p J U 3 M T Z πR iI 2 8 t M in section e E 3 0 0 0 0 0 0 1 M ) by a change of basis, with the twelve angles 3 with degeneracy 8. From this, one can find      ) and each with degeneracy four. ) and from the action of the automorphism π/ 6, each with degeneracy 4. ). Then from ( p := 4.11 we constructed an explicit example of an order three non- iπ 3 σ 2.33 4 3 with degeneracy two for each, and the matrix ( . Suppose we reduce on the 3 ˆ 4.10 T ), the corresponding O(Γ M ,..., iπ 3 , for the second one-cycle of the torus. , for the first one-cycle of the torus. 4.1 4 20), the group preserving the Minkowski metric diag(1 3.5 2 1 , p p = 1 σ σ r and exp monodromy belonging to monodromy belonging to is deduced from the action of the geometrical automorphism 2 1 iπ 3 p p 2 p and exp σ In example . 7), for J is given by eq. ( iπ 3 I 2 2 8 t 3 E 3 irπ/ which can be brought to the diagonal form ( ˆ M 7 M automorphism ˆ automorphism ˆ An order An order ˆ M In full generality, using Lemma To specify the reduction, we choose the monodromy Similarly, the order seven non-geometric automorphism wou The corresponding twisted reduction on 3 with degeneracy 16 and 4 • • π/ constants The action of ˆ another automorphism, e.g. that resulting from where 2 this basis takes the form ( surface on the vector space generated by vector space generated by by exp(2 values exp geometric automorphism thatGepner leaves point. invariant the From self-mir ( and hence the structure constants of the associatedExample gauged 7. s matrix trix This can be put in the form ( eigenvalues exp presented in subsection a change of basis. The monodromies in the non-geometric automorphisms can beten diagonalized as over elements of O(4 JHEP11(2017)084 i } } ˜ × θ 1 he 12 , and )] 1 (5.6) (5.5) p 2 In the ρ θ ,..., − 12 = ∈ {− Otherwise = 1 ) while the i . 1 p , i ε 1 θ i σ like subgroup O(20 − ( iθ moduli. About ss vector multi- ) = 1. T 1 2 × = p exp T i 2 { ε iθ on, then = O(2) 1 nkowski vacua that pre- ε ravity theory can be read tance the self-mirror sur- geometrical automorphism additional massless hyper- erated by mensional physics obtained omorphisms; the associated , each O(2) generator comes from 3 metry from KK modes cannot . 3 on for each cycle, as seen from n and to the . 20) are specified by 12 angles = 1 or , . 2 mple 3.2 1 π ε 2 π p 2 p 2 O(4 1 O(4) transformations are specified by = 2 = 2 STU supergravity at low energies, ε which satisfy exp( ε ⊂ 1 ⊂ ε i N = = θ ) conjugation. . 2 R 2 ˜ θ 2; O(2) 5.2 , O(20) such that the complex numbers × q × – 34 – , θ , 1 rotation in the space-like two-plane in the vector π 2 π p 2 times each. p 2 p O(4) q = ) after O(2 = 2 supersymmetry is preserved only if = ⊂ . This requires 1 1 2 2 θ ˜ θ ˜ p θ 2.31 N Z − , 4 20) plays a role, where the first O(2) factor acts as an order = , 2. 13 . These are then 1 monodromies, give just ˜ 2 θ > ). 7 ˜ θ ) specified by 12 angles roots of unity, . The scalar potential of the supergravity admits minima at t O(4 T p 2 , σ 4 ) = 1. p p σ 1 2 ( ˜ ⊂ θ and p 4.11 i T ˜ 2 θ and ˆ i θ 2 or 3 and − σ O(4) > 2 = 1 )]. As far as gravitini masses are concerned, only the space- ⊂ , θ p p 1 1 ρ θ θ ) generator given by eq. ( ) with ˆ Z = 2 supersymmetry as we will show below, and have three massle O( = 0 if there is no discrete torsion. If there is discrete torsi or O(2) 2; i , 2 × N ε ˜ 2.26 × θ As explained in section The parts of the monodromies in O(2) The stringy compactifications discussed in this work have Mi From these angles, the full structure of the effective superg For instance, for the order three automorphism studied in exa More explicitly, there exists a positive integer generates a finite order subgroup conjugate to a subgroup of [ = rotation in the space-like two-plane in the vector space gen 13 12 p 1 ˜ ˆ space generated by second O(2) factor acts as an order O(2) corresponding to the twothe possible corresponding choices worldsheet of description discrete in subsection torsi [O(2) p Here the O(2 As we have seen, the isometry induced by the action of the non- 5.1 Gravitini masses and supersymmetry from the asymmetric Gepner models considered in section serve same way the monodromies in fixed points of the automorphisms that reproduce the four-di are given by the primitive the angles plets S,T and U associated respectively to the axion-dilato half of theface corresponding ( asymmetric Gepner models, for ins and σ the standard form ( while in the othermultiplets, cases depending the of low-energy themoduli theory choice space contains of will some K3 be surfaces discussed and in subsection of aut off, as seen in section θ all supersymmetry is broken. Notearise that here accidental if supersym which satisfy exp( JHEP11(2017)084 , ⋆ p ]). ]): σ 8 46 (5.7) [ means that 2 ], the minima ε 2 = = 4 supergravity 1 ε rresponding Gepner . Therefore it makes N SO(4) and hence an th the Gepner model = 4 space-time super- ace-time supercharges es, which have vanish- on, ⊂ cale of the spontaneous e the remaining massless ing from the moduli space lity twists [ at pair the non-symplectic ⊂ N s of the accidental massless g on the mirror K3 surfaces phisms that we have given in variant under both automor- . rgravity framework (while ten- non-geometric constructions). 1 = 2 − p p ), i.e. such that the surface admits ζ N T 2.22 . The global structure of these moduli p · · · ⊕ σ , these moduli arise first as deformations of ] (see in particular around eq. (4.3) in [ 2 , generated by the orthogonal complement p p 8 ζ ) correspond to the intersection of the fixed- C σ T from a supergravity viewpoint. ⊗ 4.3 ⊕ ) – 35 – p p ζ 4.2 σ T ( -cyclic form ( = 2 is set by the (inverse of the) volume of the two- p T = C := ) ) p → N C 1. Then two of the gravitini remain massless, while the 1 the isometry preserves SO(3) σ ) ( ± p 2.21 ± σ = 0, all gravitini become massive and so all the spacetime T ), admits a decomposition in terms of the eigenspaces of ( = 4 = p i = 2 vacua in four dimensions. = T ε σ 2 N ( ε 2 ] and can be summarized briefly as follows (for the details see of the space-time R-symmetry is preserved. N S ε 46 R = = 1 ε 1 ε SU(4) ⊂ . One has from ( 1 R used in a particular compactification. provides precise predictions for the scalar manifolds aris p σ 3 ] and can be taken to be much smaller than the string mass scale 8 For ‘geometric’ non-symplectic automorphismsing of discrete K3 torsion surfac supersymmetry is broken. symmetry if SU(2) other two acquire an equal mass. In the worldsheet descripti The two non-geometric twists preserve the same the same choice ofmodel discrete torsion freely was acting madefrom orbifolds, for either so both the of that left-movers or the they the co both right-movers. preserve sp Whenever For a given non-geometric automorphism ˆ From the general construction of compactifications with dua We then draw the following conclusions which are in accord wi To conclude, there is a perfect agreement between the gauged • • • spaces was studied in [ The complex vector space hypermultiplets, if any, correspond tophisms K3 ˆ moduli that are in the action of the non-symplectic automorphism of our models, parametrizedhypermultiplets by discussed the in vacuum subsection expectation value of the effective four-dimensional potentialpoint ( loci of the two monodromies used in the reduction. Henc description of the vacua that was obtained in [ of the invariant lattice see Lemma the algebraic surface that preserve its sense to analyse the modeldimensional within supergravity the would four-dimensional supe be inappropriate for these 5.2 Moduli space The mathematical formulation of thesection non-geometric automor can be compatible with Then only theautomorphisms non-geometric with twists the with corresponding discrete automorphisms torsion actin th and the worldsheetsupersymmetry construction. breaking Note finally that the mass s torus [ JHEP11(2017)084 . p ζ T p one and T 3 (5.9) (5.8) \B 19 σ (5.10) (5.11) (5.12) (5.13) , T p 3 Γ as defined ok also at . −→ p p σ σ ) are given by the Z p ector space ), we expect that X, \B ( 3 and the moduli space p 2 3 surfaces, the K3 H , and the mirror K3 , : p . } φ is then given by Γ B 0 } \ has period (i.e. holomor- , γ 1 and is isomorphic to a } T p p , > ) is the holomorphic 2-form we 0 p } . For the automorphism ◦ σ } ) p − 0 X σ ⋆ > ¯ ( z σ y γ ) that induces in a natural way ) ω > 1) ◦ T p ¯ constructed from σ z, ) ( ⋆ 19 σ p (see Definition , p ¯ σ, − z , of the 21-dimensional projective space σ 3 ( σ p , p , z, σ = ( ( = ( 1 . / , ⋆ − ⋆ p p ) ) = 0 p . Then if p )) ) = 0 σ T ζ T p p es on the moduli space Γ C \B ◦ T σ σ z, z p ⊗ ( ( has period in Γ σ, σ ) = 0 ( ( Γ | ◦ T 19 p ) , , , γ 3 σ C ) ∼ = z, z · · · ⊕ Γ )) commuting with the action of )) p ( T , γ ζ p ⊗ p is the choice of an isometry 2 p – 36 – , T T σ ζ σ 16 Fix )) 19 , ( ( ) ( −→ , T p T p 3 P period point of the marked K3 surface T ) T ζ p cs σ ⊕ C (Γ T , generating a space of complex dimension five. ( ∈ ( . } P p O( M T 3 T 5 ζ z P X, marking ∈ ( K { T ∈ ) is a point of ] 2 ∈ O( σ γ X , a = = [ H ( z ,..., { ∈ { : 0 ) is called X { , T p C 2 C γ = = X which admits an action of the non-symplectic automorphism = 1 B We define also . By using the definition of ˆ φ ( { = H 3 ⊗ ) that are invariant under the action of 0 p ). ( T p , T p K ) , n 2 C T p Γ = 14 B n Ω B φ T σ p ,G H 6 ). ( 2.26 \ T p = 2 one gets a Hermitian symmetric space of complex dimension ( T σ p − T T p C ( Γ σ p W 42 φ ( T ∼ = z ) and )) lying in the following space: X n ) T y p X X { For σ ( ( ( is of complex dimension rank( ω 2. Q ω C p S 15 ], let us define − ) denotes the projective space associated with the complex v B . To understand its action on the CFT moduli space one has to lo p 46 ) = 3 ζ )) ) denotes the bilinear form on the K3 lattice Γ ). The point 2, p T X C ( ( − σ 0 , ( , P ⊗ 2 T − H 19 p > Then the K3 surface with non-symplectic automorphism To summarize, by using the description of the period map for K We now consider the non-geometric automorphism ˆ , Recall that For the self-mirror surface, for instance, local coordinat Recall that given a K3 surface 3 . In the same way as before, we define 16 15 14 T p (Γ complex ball. For phic two form gets the monomials monomial deformations of ( the subgroup of the isometry group O( and ( σ the mirror surface where a bilinear form on which is anP open (analytic) subset in a 20-dimensional quadric Following [ of marked K3 surfaces is a quotient of Ω and The moduli space of K3 surfaces with non-symplectic action b have rank( this extends in a natural way to surface with non-symplectic automorphism in section surface has period in Γ JHEP11(2017)084 = 2 (5.14) (5.15) N ): ng theory } 13 , isometries that 7 , 5 20 , , 4 3 and so there are no = 2 STU supergravity of a given prime order ) twisted by the non- 17 s of the two-torus, one ∈ { tomorphisms are given ). We have checked this p s that we have obtained ]. The new understand- p N p = 4 gauged supergravity σ e action of such an auto- 8 h a non-symplectic auto- σ 2.26 . , ( ] and found agreement. ( action of the non-geometric 2 8 N S ure ere [ li spaces, which is easier to automorphisms should apply in [ d we obtain just the as well (except naturally the = 2 ing to the Γ , , i.e. an element of the duality ice phism ts which parametrise the gauged T p 20 nsist of K3 fibrations over two-tori , is one-dimensional, and the K3 surfaces 4 der three, this is then the K3 surface in 7 \B , p > σ = 2 four-dimensional non-geometric T p 2 Γ − N so that the two automorphisms preserve , p 1 × − 2 24 p ε 20 p ( = \B p 1 – 37 – = Γ ε ∼ = ], Example 6.1, 1)). One can check that only one K3 surface Fix , 29 Fix p Σ , , this intersection is just a point 7 p Σ ˆ σ M ˆ M C and ˆ 3 dim σ is obtained by the direct product of these two spaces: p σ ) of CFTs on K3 surfaces, and hence leads to a well-defined stri 20 ). This explains why the intersection is only one point. , 4 2.26 We have analysed the models from the four-dimensional The monodromies correspond to non-geometric automorphism With two monodromy twists associated with the two one-cycle In fact the moduli space of K3 surfaces with an action by 17 , regardless of the rank of the corresponding invariant latt we have constructed provide directly the structure constan result against some of the string theory spectra computed in perspective valid at low energies. The matrices correspond compactifications of type IIA superstring theories,with that non-geometric co monodromies which lead in most cases to p p In this work we have constructed a new class of the same half of the supersymmetry). 6 Conclusion of the family admitsexample ( also a non-symplectic automorphism of or ing of these non-geometric backgroundsto in non-geometric terms automorphisms of mirrored oflattice-related Calabi-Yau aspects). three-folds STU supergravity model (provided that should consider thedo intersection case of by the case.geometric corresponding monodromies modu For ˆ instance, formassless the hypermultiplets self-mirror in K3 the surface low energy supergravity an with no hypermultiplets at low energies. and is of complex dimension (recall that we are considering h Interestingly, this dimension is the same for every automor automorphism ˆ the hypermultiplet moduli space of CFTs invariant under the of the family carry an elliptic fibration ([ by asymmetric Gepner model orbifolds, considered recently compactification. We have shown that the fixed loci of these au by pairing a non-symplecticmorphism automorphism of of the a mirrormorphism surface. K3 can surface We be have wit lifted demonstrated to that an th isometry of the lattice Γ group O(Γ JHEP11(2017)084 ring of number ]-module, ζ [ ]; the latter Z section and a 47 = 2 supersym- N nce National de la ]. Particular exam- , with the standard 2 ed supergravity con- 6 ] is called the 3 surfaces. Here, alge- e grant “New Geomet- E n K3 and the heterotic ζ [ ANR-11-IDEX-0004-02) K do, Nick Halmagyi, Jan Z C grant ST/L00044X/1. actifications [ e the scalar manifold of the , and we have showed that 2 tained, using results from the ean-Bernard Zuber for discus- l the consistency conditions of ssed further in a forthcoming ms of multiplet moduli space predic- T wists, that gives an asymmetric oduced in [ . In some cases, hypermultiplets rom the ILP LABEX (ANR-10- of constructions that correspond × et moduli space, whose dimension 3 we use the fact that thanks to the , respectively K 4 8 , E 3 = 2 supersymmetry in four dimensions, as N – 38 – ] gives a heterotic dual to our constructions consisting of a 24 [ 8 as given in Examples 4 , T = 6 3 automorphisms on the type IIA side and preserve ). Lattices with this property are very much investigated in i of order three these two lattices have the structure of a K , 3 st i σ ] for a precise introduction of the basic tools needed in this E 48 is a primitive third root of unity (recall that the ring ζ We plan to provide more details on the four-dimensional gaug The duality between the type IIA string theory compactified o Eisenstein integers theory, see [ the minima of the superpotential preserve automorphism where general introduction on the subject. toroidal reduction of the heterotic stringorbifold with construction monodromy at t fixed points;ples such of models heterotic were asymmetric intr correspond, orbifolds on are the given type by IIA CHL side, comp to symplectic automorphis expected from the string theory constructionsremain of massless such vacua in the fourmathematical dimensional literature, theories; the we corresponding have hypermultipl agrees ob with the string theory predictions. braic geometry leads us toto a non-symplectic particularly interesting class supergravities obtained from twisted reductions of bilinear form A Explicit lattice computations To get the action of the isometry on the lattice sions and correspondence. D.I.LABX-63) research supported received by support French state f funds managedand by from the ANR the ( projectRecherche. QHNS The in work of the CH program wasric supported ANR Structures by the Blanc from EPSRC String SIMI5 programm Theory” of EP/K034456/1 Age and the STF Acknowledgments We thank SamuelLouis, Boissière, Miranda Ronen Plesser, Cheng, Nathan Priddis, Alessandro Matthieu Chio Sarkis and J metry. The correspondingpublication. heterotic orbifolds will be discu struction in a companionlow-energy paper. theory in In order to particulartions show we from explicitly will algebraic that geometry analys the are hyper gauged verified, supergravity and are check met that for al these particular gaugings. string compactified on JHEP11(2017)084 - ], ]- ⊕ Q ζ [ 1 ) is 49 e Z ] in the ζ [ and by and on α, β as x Z 2 4 6 , A = E 3 matrix. The 8 on i ] is a principal E ⋆ ζ denotes the 3 i 8 (as an integral [ × , σ ]-lattices) E Z i ρ with integer coeffi- ζ [ is a set of generators asy form m × i = 6 Z i } E rators for i 4 so that T E , , and i 8 (as 6 ∈ , ζe E E 6 E 3 ) ] with an element    ζ E of 8, where θ 0 3 [ , , ζe . i Z ¯ α, β . . θ θ 3 2 8 integral lattice and on the right ). B ( 8 . , respectively 3, blocks of the form i ¯ ∈ θ 3 0 ) , ! , ]. , resp. = 6 , ζe x ζ θ 0 [ 8 ( 1    2.37 ))) = 6 bζ ]-module ⋆ ¯ i 3 θ 0 = 6 Z ] of rank 4, respectively 3. E ζ = ζ [ with its symmetric + [ , ζe , ), ( bσ

i 6 ]). Observe that the element ( Z 4 i α, β Z a ( E B E + = i . , e 32 b , respectively 3 E 2.33 , i 8 U i ! , i h ax i , e ⊕ ( E E 1 1 2 E ρ U T as a lattice over the integers. Following ([ − − i T (see [ := h , h , e i – 39 – E 6 1 1 0 ¯ E x ζ 8. Then the action of the isometry in the set of ) + e      b H · E , i {

1 θ 3 0 2 module over i ) t E to − E ¯ = i/ θ θ 3 0 0 T α, β ζb for T = 6 ( ζ 8 i ¯ θ θ 3 = 6 + i B E = ¯ = (3) θ 0 0 0 3 B , a h i st i i ( ( 2 E 3 E      E 3 1 A J h H − = 8 E h ) := and clearly is the standard lattice metric as given in the Examples ) that sends ζ 4 be a set of generators of the with the set of generators ( st e i α, β ] 4 ( i Q E ζ i [ , e E E . One can then define a bilinear form 3 Z b ¯ ζ ⊕ , e 3 2 − e ] ζ , e ζ 1 by construction does not fix any vector on [ e = Z ⋆ 3 θ . In this case the matrices of the hermitian forms have a very e as an integral lattice. One can consider a similar set of gene ⊕ σ 2 U 8 Let In a similar way one can compute the action of the automorphis Recall that the multiplication of an element With the help of MAGMA one can find a base change matrix e ] E ⊕ ζ [ Since module and a set of generators automorphism of lattice, is defined as Z ideal domain then these two lattices are free over of These are the matrices given in the equations ( generators with the standard lattice metric is given by on the lattices We call these two matrices abuse of notation we identify the bilinear form Chapter 1), consider the hermitian forms on lattice) is easy to write, since this is a block matrix with 4, where U where recall that cients such that action of the isometry in the above given set of generators considered on the left hand side as an element of the rank hand side as an element of the rank JHEP11(2017)084 , i 8 7 1 ζ B E e ⊕ := U , which ommons e . In the ]-module id ⊕ and since 4 , ζ [ U = is a basis of f Z , , } U 2 2 1 on , to ⊕ . − − 2 ⋆ U 7 e , ζe T A (2003) 054 σ (2005) 065 1 e × = ) = 1 ) = { 09 (2008) 065 ¯ ζ ζ ction of the automor- 10 2 on the basis 2 3 ]-module of rank 2 ( A A ¯ 7 ζ ⋆ 02 3 T ζ he explicit matrix of the + 3 [ σ ∈ x to get the action of the that changes the basis + (3 + 3) = ζ Z ) JHEP i ¯ 1 3 ζ JHEP redited. (3 so that , E 3 , − he previous computation that 3 1 ζ T 1 JHEP ( α, β . Here recall as given in Example e ]. , ( − 2 1 3 , f A − ))) = ))) ) = − 1 1 SPIRE e ) = , e . This is a , e IN 1 1 8 − α, β 1 fibrations e ( ][ E ( 2 3 is ζe 2 , ζe A ( 1 ⊕ A K 2 h f h ( A ζe ( U ρ b ( for a generator ρ 2 – 40 – ⊕ 1 A = (3) we have 2-hermitian matrix (to determine such forms is in e b ) + ) = ] Four-dimensional string-string-string triality U ) + 2 1 ζ 1 × [ A Z h , e α, β , ζe 8). 1 ( 1 ]). As seen above this would allow us to find the matrix , 2 e = corresponds by definition of the structure of hep-th/9508094 e ( [ A ( 2 2 2 ), which permits any use, distribution and reproduction in 48 h 2 ]. A = 6 ( A Mirrorfolds with A A ]. ]. Duality twists, orbifolds and fluxes h i 3 1 b that can then be used to give the action of ( 1 3 8 − . So we have that the automorphism sends E which is a 2 ζ SPIRE − on the lattice ⊕ , for SPIRE SPIRE 8 IN U . Now the action of the automorphism (1996) 125 3 st E i ) := IN IN ][ ⊕ st 2 ⊕ ) = CC-BY 4.0 E ][ ][ U 1 U A we get that the image of T α, β ⊕ 6= 8 we determine with MAGMA a matrix , e This article is distributed under the terms of the Creative C ( 1 , U i 1 2 = e B 459 h E A ( A Geometry for non-geometric string backgrounds ζe b 2 b 2 = 6 A A − b b i 1 of Example , e i since of the integral lattice ]-lattice we have that ⋆ 7 − E ζ 1 σ [ id hep-th/0210209 arXiv:0711.1045 hep-th/0406102 Z = ζe [ [ Nucl. Phys. [ Observe that one could use a similar method to determine the a As an example we do the explicit computations for 6= as an integral lattice. Now with 1 i e := 2 [1] C.M. Hull, [3] S. Kawai and Y. Sugawara, [4] M.J. Duff, J.T. Liu and J. Rahmfeld, [2] A. Dabholkar and C. Hull, E 2 phism to the basis forautomorphism the on standard the action basis and of we the use standard then action. this matri f to the multiplication by general a difficult problem; see [ ζ A and we get References any medium, provided the original author(s) and source are c of the base change case of denotes a primitive seventhhermitian root form of unity) but we do not know t As a with the standard bilinear form. Open Access. With the same notation as above we find that in these two cases Attribution License ( is not necessarily the caseT in general (e.g. one can check in t JHEP11(2017)084 ]. , ] 3, 2 ] , in − , SPIRE K IN (1990) 95 [ Math. 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